#7507761

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,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

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, 7414957);