#6692963

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(9,10,13,15)(11,14,12,16)(18,24)(19,28)(26,36)(27,39)(30,37)(31,43)(33,42)(45,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18,22,24)(19,29,28,40)(20,26,23,36)(21,33,35,42)(25,37,38,30)(27,32,39,34)(31,46,43,41)(44,47,48,45);(9,12,13,11)(10,14,15,16)(17,19)(18,25)(20,30)(21,31)(22,28)(23,37)(24,38)(26,40)(27,41)(29,36)(32,45)(33,48)(34,47)(35,43)(39,46)(42,44);(2,6)(5,7)(17,20,22,23)(18,26,24,36)(19,30,28,37)(21,34,35,32)(25,40,38,29)(27,42,39,33)(31,47,43,45)(41,44,46,48);(3,8)(5,7)(17,21,22,35)(18,27,24,39)(19,31,28,43)(20,32,23,34)(25,41,38,46)(26,33,36,42)(29,44,40,48)(30,45,37,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,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, 6600159);