Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,8,22,15)(7,16,20,30,21,34,19,33)(9,24,26,13,27,31,36,23)(12,25,29,18,17,35,28,32);(1,2)(3,4)(5,7)(6,12)(8,17)(9,25)(10,19)(11,21)(13,30)(14,28)(15,29)(16,31)(18,36)(20,22)(23,33)(24,34)(26,32)(27,35);(2,4)(5,8)(6,11)(7,17)(9,23)(10,14)(12,21)(13,27)(15,22)(16,32)(18,34)(19,28)(20,29)(24,36)(25,33)(26,31)(30,35)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250368, #72250378, #72250383, … |
| maximal normal subgroup of | #72250368, #72250378, #72250383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54849);