Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,17,20,28,21,27,19,12)(8,13,25,30,23,32,24,31)(18,29,22,35,33,36,34,26);(5,7,9,19,11,21,10,20)(6,12,15,17,16,28,14,27)(8,22,24,18,23,34,25,33)(13,29,30,35,32,36,31,26);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,23)(12,26)(14,31)(15,30)(16,32)(17,29)(19,34)(20,22)(21,33)(27,36)(28,35);(6,14,16,15)(8,23)(12,27,28,17)(13,30,32,31)(18,33)(22,34)(24,25)(26,29,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280601 |
| maximal normal subgroup of | #72280601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283384);