Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35166 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1458 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 298 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:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18,21,25)(16,20)(17,19)(22,24,27,28)(23,26);(5,7,11,6,9,12,8,10)(13,15,17,21)(14,18)(16,22,23,27)(19,25)(20,24)(26,28);(1,2,3,4)(6,10)(7,12)(14,19)(15,21)(16,23)(24,28);(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,24,25,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111834);