#6859126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(3,7)(4,6)(5,8)(17,18)(19,24)(20,33)(21,26)(22,36)(23,28)(25,34)(27,37)(29,44)(30,39)(31,47)(32,41)(35,43)(38,45)(40,48)(42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,32)(18,24,28,41)(20,30,21,29)(22,31,37,48)(25,42,43,38)(26,44,33,39)(27,40,36,47)(34,46,35,45);(10,14)(15,16)(17,20,23,21)(18,25,28,43)(19,29,32,30)(22,35,37,34)(24,38,41,42)(26,36,33,27)(31,46,48,45)(39,47,44,40);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,28,33)(19,30,32,29)(22,35,37,34)(24,39,41,44)(25,36,43,27)(31,46,48,45)(38,47,42,40);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,21)(25,43)(26,33)(29,30)(34,35)(38,42)(39,44)(45,46)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=hi,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, 6766322);