#6772092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#25081	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

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, 6679288);