#6724819

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1477 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)(12,15)(13,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,36)(29,45)(30,39)(31,43)(32,40)(34,48)(35,44)(37,42)(38,47)(41,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,34)(21,30)(22,32)(23,31)(25,41)(26,39)(27,40)(28,43)(29,35)(33,48)(36,46)(37,47)(38,42)(44,45);(9,11,13,12)(10,15,16,14)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,38)(26,42)(27,33)(28,44)(30,46)(31,34)(32,47)(39,48)(40,45)(41,43);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,30,32,31)(20,35,36,37)(24,39,40,43)(25,42,33,44)(29,46,47,34)(38,48,45,41);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(18,27)(19,31)(20,36)(21,23)(24,39)(29,46)(30,32)(34,47)(38,41)(40,43)(42,44)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6632015);