#7621319

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,30)(20,26)(21,37)(22,39)(23,29)(24,28)(25,33)(27,32)(31,47)(34,46)(35,45)(36,44)(38,42)(40,43)(41,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,30)(20,31,36,48)(21,22,32,24)(26,41,44,47)(27,39,37,28)(34,35,38,40)(42,45,46,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,38)(22,40)(23,36)(24,35)(25,41)(27,46)(28,45)(29,44)(30,47)(32,34)(33,48)(37,42)(39,43);(10,13)(14,16)(17,21,24,19,23,32,22,33)(18,27,28,25,29,37,39,30)(20,34,35,48,36,38,40,31)(26,42,45,47,44,46,43,41);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,39)(19,32,33,21)(20,35,36,40)(25,27,30,37)(26,43,44,45)(31,38,48,34)(41,42,47,46)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=gh,e^2=fg,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,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, 7528515);