Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3928 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)(7,11)(8,18,19,28)(9,13)(10,14)(12,20,27,21)(15,30,31,35)(16,24)(17,25)(22,29)(23,32,26,33)(34,36);(2,4)(5,7,10,17)(6,11,14,25)(8,15,19,31)(9,16,22,34)(12,26,27,23)(13,24,29,36)(18,35,28,30)(20,32,21,33);(5,8)(6,12)(7,15)(9,21)(10,19)(11,23)(13,28)(14,27)(16,33)(17,31)(18,29)(20,22)(24,35)(25,26)(30,36)(32,34);(2,4)(9,22)(13,29)(16,34)(18,28)(20,21)(24,36)(30,35)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286389, #72286818, #72286845, … |
| maximal normal subgroup of | #72286389, #72286818, #72286845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266756);