#6856507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(9,10,13,12)(11,14,16,15)(18,20)(19,26)(21,23)(24,31)(27,32)(28,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,18)(19,24)(20,22)(21,29)(23,25)(26,32)(27,28)(30,31);(3,8)(5,7)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,27)(21,26)(22,28)(23,30)(25,31)(29,32);(2,6)(5,7)(17,20,22,18)(19,27,28,24)(21,25,23,29)(26,31,30,32);(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,25,20,29)(19,26,28,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6763703);