#6767899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#23854	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)(17,18,23,27)(19,24,29,39)(20,32,35,42)(21,26,22,36)(25,37,43,34)(28,44,33,48)(30,40,31,46)(38,47,41,45);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,41)(26,40)(27,39)(28,35)(32,48)(34,47)(36,46)(37,45)(38,43)(42,44);(10,13)(14,16)(17,20,23,35)(18,25,27,43)(19,28,29,33)(21,34,22,37)(24,38,39,41)(26,42,36,32)(30,45,31,47)(40,48,46,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,39)(28,33)(30,31)(38,41)(40,46)(44,48)(45,47);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,35,37)(24,40,39,46)(25,42,43,32)(28,45,33,47)(38,48,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=hi,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

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, 6675095);