#7614730

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

1471 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,35)(30,40)(32,46)(34,42)(36,47)(37,44)(38,45)(39,41)(43,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,29)(20,34,35,30)(21,36,37,48)(23,39,38,32)(25,42,33,40)(26,43,44,47)(28,46,45,41);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,35)(18,25,27,33)(19,30,31,34)(21,32,37,39)(23,36,38,48)(24,40,29,42)(26,41,44,46)(28,43,45,47);(10,14)(13,15)(17,21,23,30,22,37,38,34)(18,26,28,40,27,44,45,42)(19,20,32,36,31,35,39,48)(24,25,41,43,29,33,46,47);(2,5)(6,8)(10,14)(13,15)(18,27)(20,35)(21,37)(24,29)(28,45)(30,34)(36,48)(41,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=h,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, 7521926);