#6875231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6782427);