#9923915

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

4298 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831111);