#8528922

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

4543 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8436118);