#7589826

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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)(19,24)(20,32)(21,37)(22,28)(23,27)(25,31)(26,36)(29,43)(30,42)(33,48)(34,40)(35,45)(38,41)(39,46)(44,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,28,25)(20,33,36,47)(21,30,23,38)(26,44,32,48)(27,41,37,42)(29,39,34,35)(40,45,43,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,25)(20,34)(21,38)(22,31)(23,30)(24,28)(26,40)(27,41)(29,36)(32,43)(33,39)(35,47)(37,42)(44,46)(45,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,26)(19,29)(21,35)(22,36)(23,39)(24,40)(25,43)(27,45)(28,32)(30,47)(31,34)(33,38)(37,46)(41,44)(42,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,30,31,38)(20,35,36,39)(24,41,25,42)(26,45,32,46)(29,47,34,33)(40,44,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7497022);