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 | |
5196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,22,29,31)(20,26,30,24)(21,25,28,32);(1,2,5,6)(3,4,7,8)(10,13)(14,15)(17,19,23,29)(18,22,27,31)(20,21,30,28)(24,32,26,25);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,30)(18,24,27,26)(19,28,29,21)(22,25,31,32);(2,6)(3,7)(9,11)(10,14)(12,16)(13,15)(17,21)(18,25)(19,20)(22,26)(23,28)(24,31)(27,32)(29,30);(1,3)(2,4)(5,7)(6,8)(17,20)(18,26)(19,21)(22,25)(23,30)(24,27)(28,29)(31,32);(1,4)(2,3)(5,8)(6,7)(17,22)(18,19)(20,25)(21,26)(23,31)(24,28)(27,29)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235763);