#7535798

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,29)(26,36)(30,38)(31,48)(32,40)(33,41)(34,42)(35,43)(37,45)(39,46)(44,47);(2,6)(7,8)(11,16)(13,15)(17,19,23,34)(18,24,29,42)(20,30,36,47)(21,33,22,32)(25,38,26,44)(27,41,28,40)(31,37,46,35)(39,43,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,29)(27,45)(28,43)(31,32)(33,46)(34,47)(39,41)(40,48)(42,44);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,26)(19,31)(21,35)(22,37)(23,36)(24,39)(25,29)(27,45)(28,43)(30,32)(33,47)(34,46)(38,41)(40,44)(42,48);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,35,36,37)(24,40,42,41)(25,43,26,45)(30,46,47,31)(38,39,44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7442994);