Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500250 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61638 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
824 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)(3,14)(4,19)(5,8)(6,12)(7,13)(9,18)(10,21)(11,24)(15,39)(16,44)(17,47)(20,53)(22,52)(23,37)(25,36)(26,30)(27,33)(28,40)(29,49)(31,60)(32,58)(34,61)(35,55)(38,50)(41,43)(42,45)(46,59)(48,62)(51,54)(56,64)(57,63);(2,8)(3,15)(5,24)(9,29)(10,19)(11,13)(12,30)(14,41)(16,20)(21,25)(26,50)(31,59)(32,33)(34,51)(35,52)(36,37)(38,40)(39,47)(42,53)(44,62)(49,58)(55,57)(60,61)(63,64);(1,3,5,17,7,15,24,43)(2,9,11,32,13,33,8,29)(4,20,21,48,23,16,25,45)(6,27,26,58,28,18,50,49)(10,34,19,51,36,31,37,59)(12,39,38,47,40,14,30,41)(22,46,55,60,56,54,57,61)(35,42,52,44,63,62,64,53);(1,4,7,23)(2,10,13,36)(3,16,15,20)(5,25,24,21)(6,22,28,56)(8,19,11,37)(9,31,33,34)(12,35,40,63)(14,42,39,62)(17,48,43,45)(18,46,27,54)(26,57,50,55)(29,59,32,51)(30,52,38,64)(41,53,47,44)(49,61,58,60)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407449);