#6508287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	#25720	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(3,8)(4,6)(5,7)(17,18,23,21)(19,22,26,24)(20,28,30,27)(25,29,32,31);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,22)(23,26)(25,29)(27,30)(31,32);(9,11)(10,13)(12,16)(14,15)(17,20,24,31,23,30,22,29)(18,25,26,28,21,32,19,27);(11,16)(13,15)(17,21,23,18)(19,22,26,24)(20,25,30,32)(27,31,28,29);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,26,21,19)(20,29,30,31)(25,28,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415483);