#6690213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597409);