Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437935 |
| 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×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,16,10,14,9,15)(7,17,20,12,21,30,19,26)(8,13,24,28,22,29,23,27)(18,31,34,25,32,36,33,35);(1,2)(3,4)(5,7)(6,12)(8,18)(9,20)(10,21)(11,19)(13,25)(14,26)(15,30)(16,17)(22,32)(23,34)(24,33)(27,36)(28,31)(29,35);(2,4)(5,7)(6,12)(8,18)(9,19)(10,21)(11,20)(13,25)(14,26)(15,17)(16,30)(22,32)(23,33)(24,34)(27,31)(28,36)(29,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,22)(11,24)(12,25)(14,29)(15,27)(16,28)(17,31)(19,33)(20,34)(21,32)(26,35)(30,36);(6,14)(8,22)(12,26)(15,16)(17,30)(18,32)(23,24)(33,34)Polycyclic presentation:
2^9|a^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312715, #72313601, #72313659, … |
| maximal normal subgroup of | #72312715, #72313601, #72313659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345136);