Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(13,14)(15,21)(16,20)(17,22)(18,23)(19,24)(25,28)(26,27);(5,6)(7,8)(13,15,17,16,19,26,18,25)(14,20,23,21,24,28,22,27);(2,4)(6,8)(13,16,17,26,19,25,18,15)(14,21,23,28,24,27,22,20);(13,17,19,18)(14,22,24,23)(15,16,26,25)(20,27,28,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72312905, #72312917, #72312973, … |
| maximal normal subgroup of | #72312905, #72312917, #72312973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265232);