Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
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×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3858 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)(7,8)(13,14,17,20,18,21,16,26)(15,19,24,27,25,28,23,22);(2,4)(9,10,11,12)(13,15,16,23,18,25,17,24)(14,19,26,22,21,28,20,27);(6,8)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,28)(25,27);(2,4)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,27,28,22)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264583, #72264604, #72264636, … |
| maximal normal subgroup of | #72264583, #72264604, #72264636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268794);