#7508536

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1423 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,34)(21,27)(22,25)(23,28)(24,33)(26,37)(30,39)(31,40)(32,44)(35,41)(36,43)(38,42)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(17,19,22,33)(18,24,25,29)(20,35,37,31)(21,32,23,30)(26,41,34,40)(27,39,28,44)(36,46,38,45)(42,47,43,48);(9,11,12,16)(10,13,14,15)(18,25)(19,30)(21,23)(24,39)(26,34)(29,44)(31,45)(32,33)(35,46)(36,38)(40,47)(41,48);(2,7)(3,8)(10,14)(13,15)(17,20,22,37)(18,26,25,34)(19,31,33,35)(21,38,23,36)(24,40,29,41)(27,43,28,42)(30,45,32,46)(39,47,44,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,28)(19,32,33,30)(20,36,37,38)(24,39,29,44)(26,42,34,43)(31,45,35,46)(40,48,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415732);