#8436585

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

4387 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,31,30)(27,33,28,32)(29,37,38,34)(35,39,36,40);(1,3)(2,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(1,4)(2,3)(5,8)(6,7)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(1,5,6,3)(2,4,8,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(1,4)(2,5)(3,8)(6,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,30,31,26)(27,33,28,32)(29,34,38,37)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=gh,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, 8343781);