#6810054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,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,g^c=gi,e^d=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, 6717250);