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 | |
4936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16);(1,2,5,4)(3,7,8,6)(18,23)(19,25)(24,30)(26,31);(1,3)(2,6)(4,7)(5,8)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,3)(2,7)(4,6)(5,8)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)Polycyclic presentation:
2^9|a^2=hi,c^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7826184);