#8436561

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4477 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343757);