#6802829

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 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)(3,7)(4,6)(5,8)(17,18,24,32,22,20,23,27)(19,28,21,26,29,25,30,31);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,29)(18,25,20,28)(21,23,30,24)(26,32,31,27);(3,8)(5,7)(10,13)(14,15)(17,20,22,18)(19,28,29,25)(21,31,30,26)(23,27,24,32);(2,6)(5,7)(17,21)(18,26)(19,23)(20,31)(22,30)(24,29)(25,32)(27,28);(9,11,12,16)(10,14,13,15)(19,29)(21,30)(25,28)(26,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6710025);