#6558440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#24265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1725 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)(19,20)(21,26)(22,27)(23,28)(24,32)(25,30)(29,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,28)(21,22)(23,30)(24,29)(26,32)(27,31);(10,13)(14,16)(17,20,22,31,23,25,24,26)(18,21,32,30,28,29,27,19);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,29)(18,26,28,31)(19,22,30,24)(20,27,25,32);(11,15)(14,16)(17,22,23,24)(18,27,28,32)(19,29,30,21)(20,31,25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465636);