Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×2−1+6 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
15667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,16,10,17,11,15)(7,13,21,29,20,30,27,31)(8,22,24,9,14,25,23,26)(18,32,33,34,28,36,35,19);(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,27)(14,28)(15,29)(16,31)(17,30)(22,32)(23,33)(24,35)(25,36)(26,34);(2,4)(5,8,10,14)(6,9,17,26)(7,18,20,28)(11,24,12,23)(13,19,30,34)(15,25,16,22)(21,35,27,33)(29,36,31,32);(5,9)(6,14)(7,19)(8,17)(10,26)(11,22)(12,25)(13,28)(15,24)(16,23)(18,30)(20,34)(21,32)(27,36)(29,35)(31,33)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72268005, #72271655, #72271686 |
| maximal normal subgroup of | #72268005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299338);