Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #412862 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2858 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15136 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29)(16,42)(17,46)(19,33)(20,40)(22,31)(23,52)(24,36)(26,37)(28,39)(30,38)(32,49)(34,45)(35,53)(41,47)(43,57)(44,58)(48,60)(50,55)(51,59)(54,56)(61,62)(63,64);(1,3)(2,9)(4,22,25,44)(5,17)(6,19)(7,18)(8,20)(10,34,21,42)(11,32)(12,40)(13,15)(14,33)(16,24,45,26)(23,51,54,62)(27,41)(28,48)(29,47)(30,49)(31,37,58,36)(35,59,50,61)(38,60)(39,46)(43,53,63,55)(52,57,56,64);(1,4,6,24,8,25,7,26)(2,10,12,36,14,21,13,37)(3,16,18,44,20,45,19,22)(5,23,28,53,30,54,29,55)(9,31,15,34,33,58,40,42)(11,35,38,52,39,50,27,56)(17,43,47,62,49,63,48,51)(32,57,41,59,46,64,60,61);(1,5,7,29,8,30,6,28)(2,11,13,27,14,39,12,38)(3,17,18,47,20,49,19,48)(4,23,26,55,25,54,24,53)(9,32,15,41,33,46,40,60)(10,35,37,56,21,50,36,52)(16,43,44,62,45,63,22,51)(31,57,34,59,58,64,42,61)Polycyclic presentation:
2^9|b^2=f,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320058);