#6459034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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

1717 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,24,28)(19,29,21,26)(20,31,30,25)(22,27,23,32);(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,23)(21,24)(22,30)(26,27)(28,31)(29,32);(2,5)(6,8)(11,15)(13,16)(17,20,24,30)(18,25,28,31)(19,22,21,23)(26,27,29,32);(11,15)(13,16)(17,21,24,19)(18,26,28,29)(20,22,30,23)(25,32,31,27);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,27,28,32)(19,30,21,20)(25,26,31,29)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6366230);