#7613310

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7520506);