#6930774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#25787	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6837970);