Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(7,10)(9,17)(12,14)(13,16)(24,32)(28,33)(30,36)(34,35);(1,2)(3,4)(5,6,11,8)(7,14,10,12)(9,13,17,16)(15,20,18,19)(22,26)(27,29)(28,33)(34,35);(5,7,11,10)(6,12,8,14)(9,18,17,15)(13,20,16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,11,6)(7,12,10,14)(9,16,17,13)(15,19,18,20);(5,9,11,17)(6,13,8,16)(7,15,10,18)(12,19,14,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,10,11,7)(6,12,8,14)(9,18,17,15)(13,19,16,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619921);