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 | #35001 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1708 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1708 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18,22,26)(16,19)(17,20)(21,25,23,27)(24,28);(5,7,11,6,9,12,8,10)(13,15,17,22)(14,18)(16,23,24,21)(19,27)(20,26)(25,28);(1,2,3,4)(6,10)(7,12)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(16,24)(19,28)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,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, 111830);