#6767917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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)(19,29)(20,35)(21,27)(22,28)(23,39)(24,33)(25,36)(26,37)(30,44)(31,41)(32,46)(34,42)(38,43)(40,48)(45,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,34)(25,40)(26,44)(27,42)(28,29)(31,37)(35,41)(36,47)(38,48)(39,46)(43,45);(10,13)(14,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,40)(26,27)(28,43)(29,45)(31,33)(32,47)(34,48)(35,39)(41,46)(42,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,41)(25,27)(28,35)(29,44)(30,33)(32,47)(34,48)(39,43)(40,46)(42,45);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,27,28,39)(19,32,33,34)(20,36,37,38)(24,42,29,46)(25,26,43,35)(30,47,31,48)(40,44,45,41)

Polycyclic presentation:

2^9|b^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6675113);