Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22,24,13)(9,16)(10,14)(11,15)(12,20)(18,32,33,31)(19,28)(21,27)(23,29,25,30)(26,34,36,35);(2,4)(5,7,10,21,11,19,9,20)(6,12,16,28,15,27,14,17)(8,18,25,35,24,33,23,34)(13,26,30,32,22,36,29,31);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32);(7,19)(12,27)(17,28)(18,33)(20,21)(26,36)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72255817, #72255842, #72255877, … |
| maximal normal subgroup of | #72255817, #72255842, #72255877, … |
| maximal quotient of | #72313313, #72313704, #72313871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260522);