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 | |
4757 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,25,26,20)(21,23,31,32)(27,30,29,28);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,29)(22,26)(23,30)(24,25)(27,31)(28,32);(2,3)(6,7)(19,26)(20,25)(27,29)(28,30);(1,2,5,3)(4,6,8,7)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32);(1,3,5,2)(4,6,8,7)(17,20)(18,19)(21,30)(22,25)(23,29)(24,26)(27,32)(28,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21,22,31)(18,23,24,32)(19,27,26,29)(20,28,25,30)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^2=h,f^2=gh,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, 7878870);