#7506785

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,33)(21,27)(22,28)(23,29)(25,36)(26,32)(30,42)(31,40)(34,48)(35,45)(37,44)(38,43)(39,47)(41,46);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,37)(25,41)(26,28)(27,40)(29,44)(30,35)(33,48)(36,46)(38,47)(39,43)(42,45);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,20,22,36)(18,25,28,33)(19,30,32,47)(21,35,23,38)(24,39,26,42)(27,43,29,45)(31,46,37,34)(40,48,44,41);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(17,19)(18,26)(20,30)(21,37)(22,32)(23,31)(24,28)(25,42)(27,40)(29,44)(33,39)(34,35)(36,47)(38,46)(41,45)(43,48);(17,21,22,23)(18,27,28,29)(19,31,32,37)(20,35,36,38)(24,40,26,44)(25,43,33,45)(30,46,47,34)(39,48,42,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=fi,d^2=f,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, 7413981);