Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99154 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1270 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(5,6)(7,11,8,10)(9,12)(13,14)(15,21)(16,19,24,26)(17,20)(18,23)(22,25,28,27);(1,2)(3,4)(5,7)(6,10)(8,9)(11,12)(13,15)(14,18,20,21)(16,22)(17,23)(19,25,26,27)(24,28);(2,4)(5,8,9,7)(6,11,12,10)(13,16)(14,19,20,26)(15,22)(17,24)(18,25,21,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=deh,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,g^b=gh,d^c=di,e^c=ei,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251431, #72251439, #72251441, … |
| maximal normal subgroup of | #72251431, #72251439, #72251441, … |
| maximal quotient of | #72255785, #72255871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28860);