Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2626 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)(5,6,10,14,11,15,9,24)(7,16,20,27,19,32,18,12)(8,13,23,30,22,28,25,29)(17,31,33,36,21,35,34,26);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,24,32)(8,21,23,34,22,17,25,33)(13,26,29,35,28,36,30,31);(2,4)(5,8)(6,13)(7,17)(9,25)(10,23)(11,22)(12,26)(14,30)(15,28)(16,31)(18,34)(19,21)(20,33)(24,29)(27,36)(32,35);(8,22)(13,28)(17,21)(23,25)(26,36)(29,30)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316817, #72316837, #72316853 |
| maximal normal subgroup of | #72316817, #72316837, #72316853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265027);