Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(10,13)(11,15)(17,18,22,21)(19,23,20,24);(9,10)(11,16)(12,13)(14,15)(17,18)(19,24)(20,23)(21,22);(1,2,4,3)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,21,24);(5,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,20,22,19)(18,23,21,24);(1,3,4,2)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72262659, #72262681, #72262741 |
| maximal normal subgroup of | #72262659, #72262681, #72262741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1402434);