Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2182 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 2182 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,23,25,19)(17,20)(21,28,24,26)(22,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,24,25,21)(17,22)(19,26,23,28)(20,27);(9,12)(13,16)(14,19)(15,21)(17,25)(18,26)(20,23)(22,24)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210088, #72240079, #72240089, … |
| maximal normal subgroup of | #72210088, #72240079, #72240089, … |
| maximal quotient of | #72210075, #72210076, #72238972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23552);