#6536134

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	#37700	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 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)(3,8)(4,6)(5,7)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,30)(25,36)(29,44)(31,43)(32,39)(34,40)(35,41)(38,46)(42,47)(45,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,28)(20,29,35,45)(21,31,23,32)(25,38,40,42)(26,39,37,43)(33,48,41,44)(34,46,36,47);(10,13)(14,16)(17,20,22,35)(18,25,27,40)(19,29,30,45)(21,36,23,34)(24,38,28,42)(26,41,37,33)(31,47,32,46)(39,44,43,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,32)(38,42)(39,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,34,35,36)(24,39,28,43)(25,33,40,41)(29,46,45,47)(38,48,42,44)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^c=ei,g^c=gi,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, 6443330);