#6468394

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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)(17,18,21,25)(19,23,26,30)(20,22,28,24)(27,29,32,31);(9,10,13,16)(11,15,12,14)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15);(9,12,13,11)(10,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,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, 6375590);