Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9986 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1360 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 28 points:
(1,2,3,4)(9,12)(13,14,18,22)(15,23,25,27)(16,19,17,21)(20,26,28,24);(5,6)(7,9)(8,10)(11,12)(14,19)(15,24)(16,17)(21,22)(23,27)(25,26);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,27)(21,23)(22,28)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72267810 |
| maximal normal subgroup of | #72267810 |
| maximal quotient of | #72267823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18675);