#6694531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=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, 6601727);