#6560114

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#38948	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(25,34)(28,44)(29,39)(30,40)(32,46)(33,43)(35,47)(36,41)(38,45)(42,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,27,40)(20,32,35,28)(21,36,22,29)(25,42,43,38)(26,41,37,39)(31,46,47,44)(33,45,34,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,34,23,35,21,33)(18,25,26,31,27,43,37,47)(19,28,29,45,30,32,36,48)(24,38,41,44,40,42,39,46);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,33,35,34)(24,39,40,41)(25,31,43,47)(28,45,32,48)(38,46,42,44);(11,15)(14,16)(19,30)(24,40)(28,32)(29,36)(38,42)(39,41)(44,46)(45,48)

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,e^d=ehi

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, 6467310);