#72311023

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	#251828	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=gh,c^a=cfg,c^b=cgh,d^a=dhi,d^b=dfgh,d^c=dhi,e^a=ehi,e^b=egh,e^c=ei,f^c=fi,f^d=fj,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,h^a=hj,h^b=hij,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311026, #72311034, #72311155, …
maximal normal subgroup of	#72311026, #72311034, #72311155, …

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