#72340944

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#6432953	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,7)(5,9)(6,12)(8,13)(11,14)(15,16)(21,35)(23,27)(33,38)(39,41);(1,3)(2,6,9,10)(4,11,13,16)(5,12)(7,15)(8,14)(17,18)(19,27)(20,31,34,24)(21,36,35,25)(22,26)(23,30)(28,41,32,39)(29,40,37,43)(33,42)(38,44)(45,47)(46,48);(1,4)(2,7)(3,11,12,16)(5,13)(6,14,10,15)(8,9)(17,19)(18,23,26,27)(20,32)(21,37,35,29)(22,30)(24,41,31,39)(25,43)(28,34)(33,45,38,46)(36,40)(42,47)(44,48);(3,12)(6,10)(11,16)(14,15)(17,20)(18,24)(19,28)(21,38)(22,34)(23,39)(25,44)(26,31)(27,41)(29,46)(30,32)(33,35)(36,42)(37,45)(40,48)(43,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,9)(10,15)(12,16)(17,21,22,35)(18,25)(19,29)(20,33,34,38)(23,40,27,43)(24,42)(26,36)(28,45)(30,37)(31,44)(32,46)(39,47,41,48)

Polycyclic presentation:

2^10|b^2=f,c^2=g,e^2=ghj,b^a=bh,c^a=cg,c^b=cfh,d^b=dghi,d^c=dj,e^a=eghij,e^b=egij,e^c=ehj,e^d=ej,f^a=fi,f^c=fi,f^d=fj,f^e=fij,g^b=gij,g^e=gj,h^b=hi,h^c=hj,h^e=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340946, #72340983, #72340997, …
maximal normal subgroup of	#72340946, #72340983, #72340997, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);