#8699211

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

4332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8606407);