#6615790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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

1467 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)(17,18,22,27,23,28,21,26)(19,29,31,39,32,42,37,24)(20,33,35,41,34,25,36,40)(30,44,47,38,45,43,46,48);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,29)(27,42)(28,39)(33,48)(34,45)(35,46)(36,47)(40,44)(41,43);(2,6)(7,8)(10,13)(11,15)(17,20,23,34)(18,25,28,33)(19,30,32,45)(21,36,22,35)(24,38,39,48)(26,41,27,40)(29,43,42,44)(31,47,37,46);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,34)(25,33)(30,45)(35,36)(38,48)(40,41)(43,44)(46,47);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,34,36)(24,29,39,42)(25,40,33,41)(30,46,45,47)(38,44,48,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6522986);