Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2535 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1478 conjugacy classes of subgroups.
| 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)(7,9)(8,11)(10,12)(13,14,17,20,18,22,16,21)(15,23,24,19,26,28,25,27);(6,9)(7,12)(10,11)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,23)(22,27);(2,4)(14,20)(16,17)(19,23)(21,22)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,d^c=di,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240262, #72240389, #72240390, … |
| maximal normal subgroup of | #72240262, #72240389, #72240390, … |
| maximal quotient of | #72257786, #72257788, #72264823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22581);