Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,18,11)(8,19)(9,22)(10,13)(12,21)(14,23)(16,29,30,35)(17,25,32,28)(20,33)(24,31,26,36)(27,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,26)(13,15)(14,25)(19,29)(20,30)(21,31)(22,28)(23,32)(24,27)(33,35)(34,36);(5,8,9,21,10,20,23,34)(6,12,14,19,13,27,22,33)(7,16,17,31,18,30,32,36)(11,24,25,35,15,26,28,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237657, #72237658, #72237672, … |
| maximal normal subgroup of | #72237657, #72237658, #72237672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31092);