#6692659

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

1429 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)(19,31)(20,26)(21,27)(22,28)(23,29)(24,30)(25,35)(32,47)(33,48)(34,44)(36,46)(37,43)(38,42)(39,41)(40,45);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,35)(18,25,29,31)(20,36,34,38)(21,33,24,39)(22,37,32,40)(26,42,44,46)(27,41,30,48)(28,45,47,43);(9,12,13,11)(10,14,15,16)(17,20,23,34)(18,26,29,44)(19,32,35,22)(21,37,24,40)(25,28,31,47)(27,43,30,45)(33,38,39,36)(41,46,48,42);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,33,35,39)(20,37,34,40)(22,36,32,38)(25,41,31,48)(26,43,44,45)(28,46,47,42);(3,8)(5,7)(17,22,23,32)(18,28,29,47)(19,34,35,20)(21,38,24,36)(25,26,31,44)(27,42,30,46)(33,37,39,40)(41,45,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 6599855);