#6501341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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

1730 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)(20,24,31,39)(21,34,36,25)(23,35,37,30)(29,43,32,46)(33,48,45,41)(38,47,40,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,28)(20,31)(23,37)(24,39)(27,42)(30,35)(33,45)(41,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,37)(20,32,31,29)(21,35,36,30)(24,40,39,38)(25,42,34,27)(33,44,45,47)(41,43,48,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,31)(18,24,26,39)(19,29,28,32)(21,33,36,45)(23,38,37,40)(25,41,34,48)(27,43,42,46)(30,44,35,47);(10,14)(13,15)(20,31)(24,39)(29,32)(33,45)(38,40)(41,48)(43,46)(44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,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, 6408537);