Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1895 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11076 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,7,14,28,11,8,12,29,41,27,40,5,26)(3,9,18,33,19,34,50,32,17,35,51,63,49,62,16,48)(4,20,24,36,25,39,56,43,22,42,57,10,30,37,23,38)(15,44,46,58,47,61,54,64,21,53,55,31,52,59,45,60);(1,3)(2,9)(4,21,22,15)(5,16)(6,18)(7,19)(8,17)(10,31,36,58)(11,32)(12,35)(13,33)(14,34)(20,44,42,53)(23,54,56,45)(24,55,57,46)(25,52,30,47)(26,48)(27,49)(28,50)(29,51)(37,59,39,61)(38,60,43,64)(40,62)(41,63);(1,4)(2,10)(3,15)(5,24)(6,23)(7,30)(8,22)(9,31)(11,39)(12,36)(13,42)(14,43)(16,46)(17,21)(18,45)(19,52)(20,41)(25,27)(26,37)(28,57)(29,56)(32,61)(33,53)(34,64)(35,58)(38,40)(44,63)(47,49)(48,59)(50,55)(51,54)(60,62)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307185, #72307193, #72307259, … |
| maximal normal subgroup of | #72307185, #72307193, #72307259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55404);