#6461725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368921);