#7508674

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(17,18,22,27)(19,28,32,24)(20,25,35,43)(21,36,23,26)(29,40,31,38)(30,41,33,39)(34,44,37,42)(45,48,46,47);(1,3)(2,6)(4,7)(5,8)(17,19,22,32)(18,24,27,28)(20,33,35,30)(21,31,23,29)(25,41,43,39)(26,38,36,40)(34,46,37,45)(42,47,44,48);(9,11,12,16)(10,13,14,15)(19,29)(21,23)(24,38)(26,36)(28,40)(30,45)(31,32)(33,46)(34,37)(39,47)(41,48)(42,44);(2,7)(3,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,34)(24,39)(26,44)(27,43)(28,41)(29,45)(31,46)(32,33)(36,42)(38,47)(40,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,27,36)(19,31,32,29)(20,34,35,37)(24,38,28,40)(25,42,43,44)(30,45,33,46)(39,48,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415870);