#7065516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972712);