#6773367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1463 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6680563);