#6555891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463087);