Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,7)(9,10,11,12)(13,14)(15,23)(16,24)(17,20)(18,22)(19,25)(21,27)(26,28);(1,2)(3,4)(5,6)(7,8)(13,15,18,19,17,24,27,28)(14,16,22,26,20,23,21,25);(2,4)(5,7,8,6)(13,16,17,23)(14,19,20,28)(15,22,24,21)(18,26,27,25)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238820, #72240061 |
| maximal normal subgroup of | #72238820, #72240061 |
| maximal quotient of | #72238831, #72240032, #72240033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30861);