#6560034

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467230);