#7590125

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fg,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497321);