#72340276

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#251827	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2777 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=gi,d^2=gi,e^2=ij,c^a=cg,c^b=cf,d^a=di,d^b=dh,d^c=di,e^a=ehij,e^b=ei,e^c=ehi,e^d=efi,f^a=fi,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^c=hj,h^d=hi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340288, #72340326, #72340364, …
maximal normal subgroup of	#72340288, #72340326, #72340364, …

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, 1, F.9);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9);;
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, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);