#7620637

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 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)(11,14)(12,13)(15,16)(17,18,22,27)(19,29,31,24)(20,25,34,41)(21,36,32,26)(23,28,37,43)(30,45,46,39)(33,44,38,40)(35,42,47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,30,34,46)(21,37,32,23)(25,39,41,45)(26,43,36,28)(33,35,38,47)(40,42,44,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,27,41)(19,30,31,46)(21,38,32,33)(23,35,37,47)(24,39,29,45)(26,44,36,40)(28,42,43,48);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,23,19,22,32,37,31)(18,26,28,24,27,36,43,29)(20,33,35,46,34,38,47,30)(25,40,42,45,41,44,48,39);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(18,27)(20,34)(24,29)(26,36)(28,43)(30,46)(33,38)(35,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fghi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=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, 7527833);