Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,21,33)(8,13)(9,24)(10,27,22,15)(12,28,31,20)(14,29)(18,19)(23,26)(25,35)(30,32)(34,36);(2,4)(5,7)(6,12)(8,18)(9,25)(10,28)(11,21)(13,30)(14,32)(15,33)(16,31)(17,27)(19,26)(20,22)(23,35)(24,36)(29,34);(1,2,3,4)(5,8,22,23,11,24,10,14)(6,13,27,29,16,9,15,26)(7,18,28,32,21,36,20,35)(12,30,33,19,31,25,17,34);(5,9,11,13)(6,14,16,23)(7,19,21,34)(8,15,24,27)(10,29,22,26)(12,18,31,36)(17,35,33,32)(20,30,28,25)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293129);