#6699994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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)(17,18,23,28)(19,24,32,40)(20,25,35,43)(21,26,22,27)(29,37,46,48)(30,38,31,39)(33,41,45,47)(34,42,36,44);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,41)(26,38)(27,39)(28,40)(29,34)(35,45)(36,46)(37,42)(43,47)(44,48);(10,14)(13,15)(17,20,23,35)(18,25,28,43)(19,29,32,46)(21,34,22,36)(24,37,40,48)(26,42,27,44)(30,45,31,33)(38,47,39,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,34,35,36)(24,38,40,39)(25,42,43,44)(29,45,46,33)(37,47,48,41);(2,5)(6,8)(11,16)(13,15)(19,31)(20,35)(21,22)(24,39)(25,43)(26,27)(29,45)(30,32)(33,46)(37,47)(38,40)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607190);