#8592222

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

4582 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8499418);