#8463720

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

4386 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,e^2=hi,f^2=ghi,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, 8370916);