#6622796

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,28,30,23)(20,32,33,41)(21,25,36,27)(24,34,40,35)(29,43,45,48)(31,39,47,38)(37,46,42,44);(2,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,29)(21,31)(22,30)(24,37)(25,38)(26,28)(27,39)(32,48)(33,45)(34,44)(35,46)(36,47)(40,42)(41,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,21,35,22,33,36,34)(18,24,27,32,26,40,25,41)(19,29,31,46,30,45,47,44)(23,37,39,48,28,42,38,43);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,30)(20,33)(23,28)(24,40)(31,47)(32,41)(34,35)(38,39);(9,12,13,11)(10,16,14,15)(17,21,22,36)(18,25,26,27)(19,31,30,47)(20,34,33,35)(23,38,28,39)(24,32,40,41)(29,44,45,46)(37,48,42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529992);