#6761312

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 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)(4,8)(5,7)(17,18,23,25,22,28,21,27)(19,24,35,40,29,39,31,37)(20,26,36,41,33,43,34,44)(30,38,46,48,32,42,45,47);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,24,28,39)(20,32,33,30)(21,31,23,35)(25,40,27,37)(26,42,43,38)(34,46,36,45)(41,47,44,48);(9,11,12,16)(10,13,14,15)(18,25)(19,29)(21,23)(24,37)(26,41)(27,28)(30,32)(34,36)(38,47)(39,40)(42,48)(43,44);(1,4,6,5)(2,7,3,8)(17,20)(18,26)(19,30)(21,34)(22,33)(23,36)(24,38)(25,41)(27,44)(28,43)(29,32)(31,45)(35,46)(37,47)(39,42)(40,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(19,29)(20,33)(24,39)(26,43)(31,35)(34,36)(37,40)(41,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6668508);