#7505170

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

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, 7412366);