#7504695

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7411891);