#9889513

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

4483 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)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26)(27,30)(28,29)(31,39)(32,34)(33,40)(35,36)(37,38);(1,2,5,7)(3,8,4,6)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,32,29)(26,30,34,28)(31,35,40,37)(33,38,39,36);(1,3,5,4)(2,6,7,8)(10,12)(11,18)(17,20)(19,24)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(1,2)(3,6)(4,8)(5,7)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,30,32,28)(26,27,34,29)(31,36,40,38)(33,37,39,35);(1,4,5,3)(2,6,7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9796709);