#6690659

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1790 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)(11,14)(12,16)(13,15)(17,18,22,23)(19,27,30,24)(20,31,34,25)(21,26,37,36)(28,38,43,32)(29,41,40,35)(33,44,47,39)(42,48,46,45);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,23,27)(20,32,34,38)(21,35,37,41)(25,28,31,43)(26,40,36,29)(33,46,47,42)(39,45,44,48);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,28)(21,33)(22,34)(23,31)(24,38)(26,39)(27,32)(29,42)(30,43)(35,48)(36,44)(37,47)(40,46)(41,45);(2,6)(5,7)(17,18,22,23)(19,24,30,27)(20,25,34,31)(21,36,37,26)(28,38,43,32)(29,41,40,35)(33,44,47,39)(42,45,46,48);(3,8)(5,7)(17,21,22,37)(18,26,23,36)(19,29,30,40)(20,33,34,47)(24,35,27,41)(25,39,31,44)(28,42,43,46)(32,45,38,48)

Polycyclic presentation:

2^9|a^2=i,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, 6597855);