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 | |
5010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,23,28,25)(20,21,27,26)(29,31,32,30);(2,6)(3,8)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,30)(27,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,10)(11,15,16,14)(18,24)(19,28)(20,27)(30,31);(1,4)(2,8)(3,6)(5,7)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,31)(27,32);(1,4)(2,3)(5,7)(6,8)(17,20)(18,26)(19,29)(21,24)(22,27)(23,30)(25,31)(28,32);(1,3)(2,7)(4,6)(5,8)(17,21,22,26)(18,27,24,20)(19,30,28,31)(23,29,25,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7884796);