Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(10,14)(11,18)(12,15)(17,23)(20,21)(22,24);(1,2)(3,4)(5,7,8,6)(9,10,15,23,13,17,12,14)(11,19,18,22,21,16,20,24);(2,4)(9,11)(10,16)(12,20)(13,21)(14,22)(15,18)(17,19)(23,24);(9,10,15,23,13,17,12,14)(11,16,18,24,21,19,20,22)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321310, #72321321, #72321379, … |
| maximal normal subgroup of | #72321310, #72321321, #72321379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272253);