#6690551

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(17,18,23,22)(19,24,29,36)(20,25,32,37)(21,33,35,26)(27,38,42,30)(28,34,43,39)(31,45,46,40)(41,44,48,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,34)(22,36)(23,29)(25,27)(26,28)(31,41)(32,38)(33,43)(35,39)(37,42)(40,47)(44,45)(46,48);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,27)(21,31)(22,37)(23,32)(24,38)(26,40)(28,41)(29,42)(30,36)(33,45)(34,44)(35,46)(39,47)(43,48);(2,6)(5,7)(17,18,23,22)(19,24,29,36)(20,25,32,37)(21,33,35,26)(27,38,42,30)(28,34,43,39)(31,45,46,40)(41,44,48,47);(3,8)(5,7)(17,21,23,35)(18,26,22,33)(19,28,29,43)(20,31,32,46)(24,39,36,34)(25,40,37,45)(27,41,42,48)(30,44,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6597747);