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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 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)(7,11,16,27)(8,18)(9,21)(10,13)(12,20)(14,24)(15,31,33,35)(17,29,22,26)(19,30)(23,28)(25,34,36,32);(1,2)(3,4)(5,7)(6,11,13,27)(8,15)(9,22)(10,16)(12,25,28,36)(14,29,21,26)(17,24)(18,35,30,31)(19,33)(20,34)(23,32);(2,4)(5,8)(6,12)(7,15)(9,23)(10,19)(11,25)(13,28)(14,30)(16,33)(17,34)(18,21)(20,24)(22,32)(26,35)(27,36)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^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, #72237515, #72237651, … |
| maximal normal subgroup of | #72237506, #72237515, #72237651, … |
| maximal quotient of | #72239457, #72279739, #72281133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30764);