#7512652

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	#35212	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,29,36)(20,24,30,38)(21,26,32,39)(27,34,33,40)(28,35,42,46)(31,37,43,47)(41,45,44,48);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(19,27)(21,32)(23,34)(26,39)(28,41)(29,33)(31,43)(35,45)(36,40)(37,47)(42,44)(46,48);(10,14)(13,15)(17,19,22,29)(18,23,25,36)(20,28,30,42)(21,33,32,27)(24,35,38,46)(26,40,39,34)(31,44,43,41)(37,48,47,45);(2,3)(6,7)(11,16)(13,15)(17,20)(18,24)(19,28)(21,31)(22,30)(23,35)(25,38)(26,37)(27,41)(29,42)(32,43)(33,44)(34,45)(36,46)(39,47)(40,48);(1,4,5,8)(2,6,3,7)(18,25)(20,30)(23,36)(26,39)(28,42)(31,43)(34,40)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,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, 7419848);