Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,21)(10,13)(11,17)(14,24)(16,32)(18,29)(19,30)(20,27)(22,26)(23,28)(25,33)(31,34)(35,36);(1,2,3,4);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,26)(14,29)(15,21)(17,24)(19,35)(20,34)(23,33)(27,36)(28,31)(30,32);(5,8,10,20)(6,12,13,27)(7,16,18,34)(9,23,24,19)(11,25,26,36)(14,30,21,28)(15,31,29,32)(17,35,22,33);(6,13)(11,26)(12,27)(14,21)(15,29)(25,36)(28,30)(31,32)Polycyclic presentation:
2^9|b^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,g^c=gi,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312064, #72312230, #72313645, … |
| maximal normal subgroup of | #72312064, #72312230, #72313645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536165);