#7513534

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,31,35,25)(20,26,21,27)(22,28,24,30)(32,47,48,40)(33,45,38,41)(34,43,36,42)(37,44,39,46);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,23,35)(18,25,29,31)(20,33,21,38)(22,36,24,34)(26,41,27,45)(28,43,30,42)(32,39,48,37)(40,46,47,44);(10,13)(14,16)(17,20,23,21)(18,26,29,27)(19,32,35,48)(22,37,24,39)(25,40,31,47)(28,44,30,46)(33,36,38,34)(41,43,45,42);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,26)(19,33,35,38)(22,37,24,39)(25,41,31,45)(28,44,30,46)(32,36,48,34)(40,43,47,42);(1,5,6,4)(2,3,7,8)(17,22,23,24)(18,28,29,30)(19,34,35,36)(20,37,21,39)(25,42,31,43)(26,44,27,46)(32,33,48,38)(40,41,47,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,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, 7420730);