#6560718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(9,10,14,13)(11,16,12,15)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(31,46)(34,40)(35,41)(37,43)(44,47)(45,48);(1,3,5,4)(2,6,7,8)(9,11,14,12)(10,15,13,16)(17,19,23,30)(18,24,27,39)(20,28,34,44)(21,29,22,31)(25,37,41,48)(26,38,36,46)(32,42,40,47)(33,43,35,45);(1,4,5,3)(2,6,7,8)(9,10,14,13)(11,16,12,15)(17,20,22,33,23,34,21,35)(18,25,26,32,27,41,36,40)(19,28,31,43,30,44,29,45)(24,37,38,42,39,48,46,47);(9,12,14,11)(10,15,13,16)(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,34,35)(24,38,39,46)(25,40,41,32)(28,43,44,45)(37,47,48,42);(9,13,14,10)(11,16,12,15)(19,30)(24,39)(28,44)(29,31)(37,48)(38,46)(42,47)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467914);