#8436281

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

4408 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)(10,16)(11,12)(18,23)(19,20)(26,29)(27,35)(28,38)(34,40);(3,7)(6,8)(9,10)(11,17)(12,14)(13,18)(15,16)(19,24)(20,21)(22,23)(25,26,31,29)(27,36,35,32)(28,39,38,33)(30,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,14)(13,19,22,20)(18,24,23,21)(25,27)(26,32)(28,37)(29,36)(30,38)(31,35)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,16,14)(13,20,22,19)(18,24,23,21)(25,28,31,38)(26,33,29,39)(27,37,35,30)(32,40,36,34);(1,2)(3,6)(4,5)(7,8)(9,13,15,22)(10,18,16,23)(11,19,12,20)(14,21,17,24)(25,29,31,26)(27,36,35,32)(28,39,38,33)(30,40,37,34);(9,14,15,17)(10,12,16,11)(13,21,22,24)(18,20,23,19)(25,30,31,37)(26,34,29,40)(27,38,35,28)(32,39,36,33)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=h,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, 8343477);