Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32);(2,4)(6,8)(9,10,13,11)(12,14,16,15)(21,29)(24,31)(27,28)(30,32);(3,7)(6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32);(1,4,5,2)(3,8,7,6)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232716);