Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3456 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)(5,6,9,13,10,14,20,26)(7,11,17,23,16,22,30,34)(8,12,19,25,18,24,31,35)(15,27,29,36,28,21,32,33);(5,7,9,17,10,16,20,30)(6,11,13,23,14,22,26,34)(8,15,19,29,18,28,31,32)(12,21,25,33,24,27,35,36);(5,8)(6,12)(7,15)(9,19)(10,18)(11,21)(13,25)(14,24)(16,28)(17,29)(20,31)(22,27)(23,33)(26,35)(30,32)(34,36);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^b=fi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250020, #72256961, #72257140, … |
| maximal normal subgroup of | #72250020, #72256961, #72257140, … |
| maximal quotient of | #72317356, #72317360, C2×#392605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299801);