#6766582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6673778);