#8436286

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gh,e^2=g,f^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, 8343482);