#9909805

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

4463 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)(4,6)(5,8)(9,10)(11,16)(12,14)(13,18)(15,17)(19,21)(20,23)(22,24)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,40,36,34);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,21,24,23)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,34,38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,12)(18,24)(19,20)(25,28,31,39)(26,33,29,38)(27,30,37,36)(32,40,35,34);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,24,21)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,34)(35,40);(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,21)(17,24)(25,29,31,26)(27,32,37,35)(28,38,39,33)(30,40,36,34);(9,14)(10,11)(12,17)(13,21)(15,16)(18,20)(19,24)(22,23)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,38,35,33)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gi,e^2=ghi,f^2=ghi,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, 9817001);