#6917352

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884937
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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)(10,14)(12,19)(15,21)(17,20)(25,26)(27,31)(28,35)(29,38)(30,33)(32,34)(36,40)(37,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,35,40,39);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,37)(26,32,33,40)(27,34,29,36)(31,39,38,35);(1,4,5,3)(2,6,7,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(28,37)(32,40)(34,36)(35,39);(10,14)(15,21)(16,22)(23,24)(25,27,30,29)(26,31,33,38)(28,34,37,36)(32,39,40,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,g^c=gi

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, 6824548);