Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 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)(5,6,10,18,12,17,11,16)(7,19,22,13,24,33,23,34)(8,25,26,35,21,32,27,14)(9,28,29,31,30,36,20,15);(5,7,10,22,12,24,11,23)(6,13,16,19,17,34,18,33)(8,20,26,9,21,29,27,30)(14,15,32,36,35,31,25,28);(5,8)(6,14)(7,20)(9,22)(10,26)(11,27)(12,21)(13,31)(15,34)(16,32)(17,35)(18,25)(19,28)(23,30)(24,29)(33,36);(2,4)(5,9,10,29,12,30,11,20)(6,15,16,36,17,31,18,28)(7,21,22,27,24,8,23,26)(13,32,19,35,34,25,33,14)Polycyclic presentation:
2^9|a^2=eg,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308196 |
| maximal normal subgroup of | #72308196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309137);