Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11121 |
| 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 | |
667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,14,9,15,8,33)(3,16,20,49,19,46,32,50)(4,11,23,42,24,44,34,43)(5,25,27,13,6,29,28,12)(10,35,38,18,37,57,39,30)(17,51,54,62,53,48,55,40)(21,58,56,36,52,64,59,63)(22,45,26,41,60,47,31,61);(1,3,5,18,8,20,28,57,9,19,6,30,7,32,27,35)(2,10,12,16,14,39,25,50,15,37,13,46,33,38,29,49)(4,21,22,54,23,59,26,17,24,52,60,55,34,56,31,53)(11,40,41,58,43,51,45,36,44,62,61,64,42,48,47,63);(1,4)(2,11)(3,17)(5,26)(6,31)(7,23)(8,34)(9,24)(10,36)(12,45)(13,47)(14,42)(15,44)(16,48)(18,56)(19,53)(20,54)(21,35)(22,28)(25,41)(27,60)(29,61)(30,59)(32,55)(33,43)(37,63)(38,64)(39,58)(40,49)(46,51)(50,62)(52,57)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307184, #72307228, #72307275, … |
| maximal normal subgroup of | #72307184, #72307228, #72307275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55564);