#6855539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1267 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)(9,10,13,12)(11,14,16,15)(17,18,22,28)(19,30,33,24)(20,35,38,26)(21,29,23,27)(25,39,45,37)(31,41,36,46)(32,42,34,43)(40,47,44,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19,22,33)(18,24,28,30)(20,36,38,31)(21,32,23,34)(25,44,45,40)(26,46,35,41)(27,42,29,43)(37,48,39,47);(9,10,13,12)(11,15,16,14)(17,20)(18,25)(19,31)(21,37)(22,38)(23,39)(24,40)(26,29)(27,35)(28,45)(30,44)(32,47)(33,36)(34,48)(41,43)(42,46);(1,3,4,7)(2,6,5,8)(17,20)(18,26)(19,31)(21,39)(22,38)(23,37)(24,41)(25,29)(27,45)(28,35)(30,46)(32,48)(33,36)(34,47)(40,43)(42,44);(2,5)(6,8)(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,37,38,39)(24,42,30,43)(25,35,45,26)(31,47,36,48)(40,46,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6762735);