Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,10,11,9,8,7,12)(13,14,15,18,17,22,23,21)(16,25,24,20,27,19,26,28);(1,2,3,4)(5,7,9,10)(6,11,8,12)(13,15,17,23)(14,18,22,21)(16,24,27,26)(19,28,25,20);(5,8)(6,9)(7,11)(10,12)(14,19)(15,23)(16,26)(18,20)(21,28)(22,25)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=di,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318762, #72318766, #72318781, … |
| maximal normal subgroup of | #72318762, #72318766, #72318781, … |
| maximal quotient of | #72322472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41605);