Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1999 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,23)(9,13)(10,27)(11,16)(12,19)(14,25)(15,24)(18,20)(21,32)(22,29)(26,31)(28,35)(30,36)(33,34);(5,7,9,21,10,22,25,35,11,20,26,36,24,34,8,19)(6,12,23,33,15,30,31,18,16,28,14,29,27,32,13,17);(5,6)(7,18,20,17)(8,23)(9,13)(10,27)(11,16)(12,19,28,35)(14,25)(15,24)(21,30,36,32)(22,33,34,29)(26,31)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322684, #72322695, #72322696, … |
| maximal normal subgroup of | #72322684, #72322695, #72322696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56452);