#6442147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ 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)(11,13)(12,14)(15,16)(17,18)(19,20)(21,31)(22,26)(23,25)(24,29)(27,32)(28,30);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18,22,26)(19,27,29,30)(20,28,24,32)(21,25,23,31);(10,14)(13,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,30)(27,31);(17,20,22,24)(18,19,26,29)(21,30,23,27)(25,32,31,28);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,26,31)(19,28,29,32)(20,27,24,30)

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,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, 6349343);