#7508471

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(19,27)(20,32)(21,24)(22,26)(23,30)(25,34)(28,37)(29,38)(31,42)(33,40)(35,45)(36,39)(41,46)(43,48)(44,47);(1,3)(2,6)(4,7)(5,8)(17,19,21,30)(18,23,24,27)(20,33,34,29)(22,28,36,31)(25,40,32,38)(26,42,39,37)(35,44,46,43)(41,47,45,48);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(22,36)(23,37)(25,32)(27,42)(29,43)(30,31)(33,44)(35,46)(38,47)(40,48);(2,7)(3,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,32)(26,41)(27,40)(28,43)(30,33)(31,44)(36,46)(37,47)(39,45)(42,48);(1,4,5,6)(2,3,7,8)(20,34)(25,32)(29,33)(35,46)(38,40)(41,45)(43,44)(47,48)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7415667);