Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2578 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2578 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14)(15,21)(16,24,25,19)(17,20)(18,23)(22,26,28,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18,20,21)(16,22)(17,23)(19,26,24,27)(25,28);(2,4)(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,f^b=fh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219460, #72241167, #72241183, … |
| maximal normal subgroup of | #72219460, #72241167, #72241183, … |
| maximal quotient of | #72219423, #72219424, #72226349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13004);