#6692667

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(17,18)(19,25)(20,33)(21,37)(22,28)(23,29)(24,27)(26,36)(30,48)(31,43)(32,42)(34,47)(35,45)(38,44)(39,41)(40,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,25,29,42)(20,34,36,38)(21,31,24,39)(22,35,30,40)(26,44,33,47)(27,41,37,43)(28,45,48,46);(9,12,13,11)(10,14,15,16)(17,20)(18,26)(19,30)(21,35)(22,32)(23,36)(24,40)(25,28)(27,45)(29,33)(31,38)(34,39)(37,46)(41,44)(42,48)(43,47);(2,6)(5,7)(17,21,23,24)(18,27,29,37)(19,31,32,39)(20,35,36,40)(22,34,30,38)(25,41,42,43)(26,45,33,46)(28,44,48,47);(3,8)(5,7)(17,22,23,30)(18,28,29,48)(19,20,32,36)(21,38,24,34)(25,33,42,26)(27,47,37,44)(31,40,39,35)(41,45,43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 6599863);