#6871528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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)(10,13)(11,15)(17,18,23,36,22,26,21,27)(19,24,37,48,29,39,30,40)(20,31,35,25,34,43,33,42)(28,44,46,38,32,47,45,41);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,32)(21,30)(22,29)(23,37)(25,41)(26,39)(27,40)(28,34)(31,47)(33,46)(35,45)(36,48)(38,42)(43,44);(10,13)(14,16)(17,20,22,34)(18,25,26,42)(19,28,29,32)(21,35,23,33)(24,38,39,41)(27,43,36,31)(30,46,37,45)(40,47,48,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,26)(19,29)(25,42)(27,36)(28,32)(30,37)(31,43)(45,46);(1,3,4,7)(2,6,5,8)(19,29)(24,39)(28,32)(30,37)(38,41)(40,48)(44,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6778724);