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 | |
5074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,13)(14,16)(17,18,23,25)(19,24,20,26)(21,32,27,22)(28,31,30,29);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,28)(22,29)(25,26)(27,30)(31,32);(2,3)(6,7)(9,11,12,15)(10,14,13,16)(18,25)(19,20)(21,27)(29,31);(1,3,5,2)(4,7,8,6)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29);(1,4,5,8)(2,6,3,7)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31);(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620399);