Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,18,11)(8,12)(9,13)(10,14)(15,22)(17,23,30,26)(19,25,32,24)(20,28)(21,29)(27,33)(31,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,17)(9,19)(10,18)(12,26)(13,24)(14,16)(15,25)(20,30)(21,31)(22,32)(23,28)(27,35)(29,34)(33,36);(2,4)(5,8)(6,12)(7,17)(9,21)(10,20)(11,23)(13,29)(14,28)(15,27)(16,26)(18,30)(19,31)(22,33)(24,35)(25,34)(32,36);(6,13,14,15)(8,20)(11,24,16,25)(12,27,28,29)(17,30)(21,33)(23,34,26,35)(31,36)Polycyclic presentation:
2^9|a^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285577, #72285613, #72285668, … |
| maximal normal subgroup of | #72285577, #72285613, #72285668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303293);