#9726508

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

4172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9633704);