#6520684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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

1264 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:

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

Polycyclic presentation:

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

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, 6427880);