#7013906

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

1270 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,e^2=fi,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, 6921102);