Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4102 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)(5,6,9,14,10,13,21,27)(7,11,17,23,16,24,20,28)(8,12,19,26,18,25,31,35)(15,22,30,34,29,33,32,36);(5,7)(6,11,13,24)(8,15)(9,20)(10,16)(12,22,25,33)(14,28,27,23)(17,21)(18,29)(19,32)(26,36,35,34)(30,31);(2,4)(5,8)(6,12)(7,15)(9,19)(10,18)(11,22)(13,25)(14,26)(16,29)(17,30)(20,32)(21,31)(23,34)(24,33)(27,35)(28,36);(8,18)(12,25)(15,29)(19,31)(22,33)(26,35)(30,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72303291, #72303311, #72303325, … |
| maximal normal subgroup of | #72303291, #72303311, #72303325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260610);