Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2478 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)(7,12,18,27)(8,20)(9,16)(10,15)(11,14)(13,22)(17,31,33,35)(19,29,23,26)(21,30)(24,28)(25,34,36,32);(1,2)(3,4)(5,7,10,19,11,18,9,23)(6,12,16,26,14,27,15,29)(8,17,22,34,21,33,24,32)(13,25,20,35,28,36,30,31);(2,4)(5,8)(6,13)(7,17)(9,24)(10,22)(11,21)(12,25)(14,28)(15,30)(16,20)(18,33)(19,34)(23,32)(26,35)(27,36)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237506, #72237508, #72239774, … |
| maximal normal subgroup of | #72237506, #72237508, #72239774, … |
| maximal quotient of | #72239756, #72239762, #72239764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30765);