#7511842

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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

1610 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)(19,25)(20,31)(21,26)(22,27)(23,29)(24,33)(28,38)(30,43)(32,40)(34,45)(35,44)(36,39)(37,42)(41,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,23,26,25)(20,32,33,47)(22,36,35,30)(24,40,31,48)(27,43,44,39)(28,37,46,34)(38,45,41,42);(10,14)(13,15)(17,20,21,33)(18,24,26,31)(19,28,29,46)(22,37,35,34)(23,38,25,41)(27,45,44,42)(30,47,36,32)(39,48,43,40);(2,7)(3,8)(11,16)(13,15)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,41)(27,43)(31,38)(32,34)(33,46)(35,36)(37,47)(39,44)(40,45)(42,48);(1,4,5,6)(2,3,7,8)(19,29)(23,25)(28,46)(30,36)(32,47)(38,41)(39,43)(40,48)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419038);