#6461290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,26,21,24,20,27)(19,25,28,23,30,32,29,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,26)(27,32);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(18,24)(23,31)(25,32)(26,27);(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,26)(24,32)(27,31);(17,20,21,22)(18,26,24,27)(19,28,30,29)(23,25,31,32)

Polycyclic presentation:

2^9|a^2=fg,c^2=ghi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6368486);