#7014395

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6921591);