Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10712 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,14,28,42,8,13,29,41,5,11,26,39)(3,15,19,44,20,32,52,59,17,9,30,33,18,34,50,61)(4,21,23,54,22,37,55,40,24,38,27,56,25,10,35,36)(16,45,48,63,47,31,57,58,46,53,51,64,49,43,62,60);(1,3)(2,9,13,15)(4,16)(5,18)(6,19)(7,20)(8,17)(10,31,37,43)(11,32,14,34)(12,33,41,44)(21,53,38,45)(22,47)(23,48)(24,46)(25,49)(26,50)(27,51)(28,52)(29,30)(35,62)(36,58,40,60)(39,59,42,61)(54,64,56,63)(55,57);(1,4)(2,10)(3,16)(5,25)(6,27)(7,22)(8,24)(9,31)(11,38)(12,40)(13,37)(14,21)(15,43)(17,46)(18,49)(19,51)(20,47)(23,29)(26,55)(28,35)(30,48)(32,45)(33,60)(34,53)(36,41)(39,54)(42,56)(44,58)(50,57)(52,62)(59,64)(61,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,d^b=di,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281050, #72281086, #72281094, … |
| maximal normal subgroup of | #72281050, #72281086, #72281094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53584);