Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
662 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)(7,18,21,35)(8,23,27,33)(9,24)(10,16)(11,17)(12,31)(13,14,26,32)(15,30)(19,29)(20,25,22,36)(28,34);(1,2)(3,4)(5,7,9,14,12,22,30,33,11,21,29,32,10,20,28,23)(6,13,15,18,16,8,24,25,17,26,34,35,31,27,19,36);(2,4)(5,8)(6,14)(7,19)(9,25)(10,13)(11,27)(12,26)(15,22)(16,33)(17,32)(18,28)(20,34)(21,24)(23,31)(29,36)(30,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250568, #72250570, #72250583, … |
| maximal normal subgroup of | #72250568, #72250570, #72250583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55482);