#6817243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724439);