#8612239

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

4594 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)(11,19)(12,22)(13,20)(14,16)(15,21)(17,23)(18,24)(25,26,32,35)(27,36,38,33)(28,31,29,39)(30,34,40,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,22)(14,21)(16,19)(17,24)(20,23)(25,27,32,38)(26,33,35,36)(28,34,29,37)(30,39,40,31);(2,3)(6,7)(10,16)(13,24)(15,19)(17,22)(25,28,32,29)(26,31,35,39)(27,34,38,37)(30,36,40,33);(4,8)(6,7)(9,12,14,23)(10,17,16,22)(11,20,21,18)(13,19,24,15)(25,29,32,28)(26,31,35,39)(27,37,38,34)(30,36,40,33);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,17)(12,19)(14,24)(15,23)(16,20)(21,22)(25,30)(26,34)(27,31)(28,33)(29,36)(32,40)(35,37)(38,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,19)(12,18)(13,17)(14,21)(15,16)(20,23)(22,24)(25,31)(26,29)(27,30)(28,35)(32,39)(33,37)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=i,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, 8519435);