#9909806

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

4305 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)(9,10,15,17)(11,16,19,14)(12,13,18,24)(20,21,23,22)(26,30)(28,36)(31,33)(34,40);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,20)(13,21)(14,17)(15,19)(18,23)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,19)(12,18)(21,22)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,18)(10,13,17,24)(11,20,19,23)(14,22,16,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,23)(19,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,14)(10,19)(11,17)(12,22)(13,23)(15,16)(18,21)(20,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 9817002);