Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3530 conjugacy classes of subgroups; only the 618 normal subgroups are shown.
This group has 3530 conjugacy classes of subgroups — too many to draw. Only its 618 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(5,7)(6,9)(8,12)(10,11)(16,24)(19,26)(22,28)(25,27);(1,2,3,4)(7,12)(9,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(6,10)(9,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284354, #72284560, #72284802, … |
| maximal normal subgroup of | #72284354, #72284560, #72284802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263113);