#6455034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,24,29,31)(20,25,30,32)(21,26,22,27);(1,3,4,8)(2,5,6,7)(19,20)(21,22)(24,25)(26,27)(29,30)(31,32);(10,13)(14,16)(17,19)(18,24)(20,21)(22,30)(23,29)(25,26)(27,32)(28,31);(17,20,23,30)(18,25,28,32)(19,21,29,22)(24,26,31,27);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,28,27)(19,20,29,30)(24,25,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6362230);