Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14338 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,12)(3,13,16,22,15,27,20,8)(4,9,18,25,19,26,21,28)(14,29,30,24,17,32,31,23);(8,22)(13,27)(23,24)(29,32);(1,3,6,16,7,15,5,20)(2,8,12,13,11,22,10,27)(4,17,18,30,19,14,21,31)(9,24,25,29,26,23,28,32);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,23)(10,25)(11,26)(12,28)(13,29)(15,17)(16,30)(20,31)(22,24)(27,32)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286021, #72286119, #72286225, … |
| maximal normal subgroup of | #72286021, #72286119, #72286225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289747);