#8924760

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

4292 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831956);