#6802653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709849);