#8682297

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

4685 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8589493);