#6467847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,17)(13,18)(14,20)(15,21)(16,24)(19,30)(23,33)(25,32)(26,39)(27,34)(28,35)(29,37)(31,38)(36,40);(9,11,14,24,15,18,30,25)(10,16,19,22,21,32,20,13)(12,26,29,23,28,38,40,34)(17,33,36,39,35,27,37,31);(2,5)(6,8)(9,12,14,29,15,28,30,40)(10,17,20,37,21,35,19,36)(11,23,25,26,18,34,24,38)(13,27,16,31,22,33,32,39);(9,13,14,16,15,22,30,32)(10,18,20,24,21,11,19,25)(12,27,29,31,28,33,40,39)(17,34,37,38,35,23,36,26);(1,3,4,7)(2,6,5,8)(12,28)(17,35)(23,34)(26,38)(27,33)(29,40)(31,39)(36,37)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375043);