#6508322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	#25731	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415518);