Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1950 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)(5,6)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26)(27,28);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(22,25)(26,27);(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^d=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72257789, #72257949, #72315983, … |
| maximal normal subgroup of | #72257789, #72257949, #72315983, … |
| maximal quotient of | #72264495 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299535);