#7620916

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

1280 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)(17,18,22,27)(19,29,31,24)(20,25,34,39)(21,26,32,41)(23,28,36,42)(30,43,44,37)(33,38,45,47)(35,40,46,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,30,34,44)(21,36,32,23)(25,37,39,43)(26,28,41,42)(33,46,45,35)(38,40,47,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,27,39)(19,30,31,44)(21,33,32,45)(23,35,36,46)(24,37,29,43)(26,38,41,47)(28,40,42,48);(10,13)(14,16)(17,21,23,19,22,32,36,31)(18,26,28,29,27,41,42,24)(20,33,35,30,34,45,46,44)(25,38,40,43,39,47,48,37);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,27)(20,34)(24,29)(26,41)(28,42)(30,44)(33,45)(35,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=gh,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7528112);