#6718233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 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)(18,23)(19,27)(25,31)(26,35)(29,42)(30,36)(40,41)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,28,27,39)(20,31,32,25)(22,35,34,26)(24,36,37,30)(29,44,42,43)(33,41,47,40)(38,45,46,48);(10,15)(14,16)(17,19,21,27)(18,24,23,37)(20,29,32,42)(22,36,34,30)(25,38,31,46)(26,39,35,28)(33,48,47,45)(40,43,41,44);(2,6)(5,7)(17,20)(18,25)(19,29)(21,32)(22,33)(23,31)(24,38)(26,41)(27,42)(28,43)(30,45)(34,47)(35,40)(36,48)(37,46)(39,44);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(18,26)(20,32)(22,34)(23,35)(24,39)(25,40)(28,37)(29,42)(30,36)(31,41)(38,43)(44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6625429);