#6573336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(9,10,15,14)(11,12,13,16)(17,18,23,26,22,28,21,27)(19,29,31,24,33,42,32,38)(20,25,36,40,34,39,35,41)(30,43,45,37,44,48,46,47);(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,37)(26,29)(27,42)(28,38)(34,44)(35,45)(36,46)(39,47)(40,43)(41,48);(2,6)(7,8)(9,12,15,16)(10,11,14,13)(17,20,22,34)(18,25,28,39)(19,30,33,44)(21,35,23,36)(24,37,38,47)(26,40,27,41)(29,43,42,48)(31,45,32,46);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,12)(20,34)(25,39)(30,44)(35,36)(37,47)(40,41)(43,48)(45,46);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,12,13,16)(17,21,22,23)(18,26,28,27)(19,31,33,32)(20,35,34,36)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480532);