#6802644

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(9,10)(11,15)(12,14)(13,16)(17,18,21,25,22,26,35,41)(19,27,29,37,30,40,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(2,5)(3,7)(9,11,12,16)(10,13,14,15);(11,16)(13,15)(17,19,22,30)(18,23,26,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(25,40,41,27)(33,48,38,45)(39,43,47,42);(2,5)(6,8)(10,14)(13,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,39)(26,31)(27,42)(29,45)(30,44)(34,48)(35,38)(37,46)(40,43)(41,47);(1,3,4,7)(2,6,5,8)(20,32)(24,31)(28,44)(33,38)(36,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=fh,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, 6709840);