#8891728

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

4603 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798924);