Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21790 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,20,25,7,9,19,24,4,17,18,30)(3,12,15,27,16,22,29,31,14,26,28,32,13,8,21,23);(1,3,4,13,7,14,6,16)(2,8,10,12,9,22,17,26)(5,15,18,21,19,28,20,29)(11,23,25,27,24,31,30,32);(8,22)(12,26)(23,31)(27,32);(2,9)(5,19)(8,22)(10,17)(12,26)(15,28)(18,20)(21,29)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292716, #72292724, C2×#496304 |
| maximal normal subgroup of | #72292716, #72292724, C2×#496304 |
| maximal quotient of | C2×#496304, #72292826 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403500);