#7526914

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,21,25)(19,27,29,23)(20,31,33,24)(22,26,35,42)(28,38,46,45)(30,43,36,39)(32,40,47,48)(34,44,37,41);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,29)(18,23,25,27)(20,32,33,47)(22,36,35,30)(24,40,31,48)(26,43,42,39)(28,37,46,34)(38,44,45,41);(10,14)(11,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,38)(25,27)(26,39)(31,45)(32,34)(33,46)(35,36)(37,47)(40,41)(42,43)(44,48);(10,14)(13,16)(17,20,21,33)(18,24,25,31)(19,28,29,46)(22,37,35,34)(23,38,27,45)(26,44,42,41)(30,47,36,32)(39,48,43,40);(1,3,4,8)(2,6,5,7)(18,25)(19,29)(24,31)(26,42)(28,46)(30,36)(32,47)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434110);