Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2406 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,21,23,28)(16,20)(17,19)(18,22,26,24)(25,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15)(14,18)(16,24)(17,23)(19,26)(20,28)(21,27)(22,25);(7,12)(9,11)(14,19)(16,25)(18,26)(22,24)Polycyclic presentation:
2^9|a^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234968, #72261158, #72261175 |
| maximal normal subgroup of | #72234968, #72261158, #72261175 |
| maximal quotient of | #72237000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356729);