#6563644

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(19,28)(20,33)(21,26)(22,36)(23,27)(24,31)(25,34)(29,38)(30,40)(32,39)(35,41)(37,42)(43,44)(45,47)(46,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,38)(26,39)(27,40)(28,36)(33,43)(34,45)(35,46)(37,44)(41,47)(42,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,43)(30,44)(31,45)(32,46)(33,36)(39,47)(40,48);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,35,37)(24,28,40,39)(25,41,42,33)(29,44,46,45)(38,43,48,47);(11,15)(14,16)(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,37)(24,39,40,28)(25,41,42,33)(29,45,46,44)(38,47,48,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=fgi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6470840);