#6767488

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	#25133	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(18,22)(19,23)(20,27)(25,31)(26,29)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,26,29)(20,28,27,24)(25,31,32,30);(10,13)(14,16)(17,19,21,26)(18,23,22,29)(20,25,27,32)(24,30,28,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,32)(29,31);(3,8)(5,7)(10,13)(14,16)(19,26)(20,27)(23,29)(24,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6674684);