#7611251

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518447);