#6506987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#25131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,35,25)(23,36,37,30)(29,43,31,46)(33,48,44,41)(38,47,39,45);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,42)(29,31)(30,36)(38,39)(43,46)(45,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,28,35,36)(18,23,25,27,26,37,34,42)(20,31,33,47,32,29,44,45)(24,39,41,46,40,38,48,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,32)(18,24,26,40)(19,29,28,31)(21,33,35,44)(23,38,37,39)(25,41,34,48)(27,43,42,46)(30,45,36,47);(10,14)(13,15)(19,28)(20,32)(23,37)(24,40)(27,42)(30,36)(33,44)(41,48)

Polycyclic presentation:

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

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, 6414183);