Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #759288 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(10,13)(11,15)(18,22)(20,24);(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,20,23,24);(5,6)(7,8)(11,15)(14,16)(18,22)(20,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,20,22,24);(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|b^2=h,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232084, #72232113, #72232141, … |
| maximal normal subgroup of | #72232084, #72232113, #72232141, … |
| maximal quotient of | #72237643, #72250268, #72293954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 667568);