Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13832 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,30)(3,9,18,34,16,35,17,33)(4,19,24,10,22,38,23,39)(5,25,28,40,29,41,27,11)(14,42,20,50,45,36,44,31)(15,46,47,58,26,55,48,32)(21,37,53,61,52,62,54,49)(43,63,51,59,56,57,64,60);(1,3,7,18,8,16,6,17)(2,9,13,33,12,35,30,34)(4,20,24,14,22,44,23,45)(5,26,28,47,29,15,27,48)(10,36,19,31,39,42,38,50)(11,32,41,46,40,58,25,55)(21,51,54,56,52,64,53,43)(37,60,61,57,62,59,49,63);(1,4,6,23,8,22,7,24)(2,10,30,38,12,39,13,19)(3,14,17,20,16,45,18,44)(5,21,28,54,29,52,27,53)(9,31,34,36,35,50,33,42)(11,37,41,61,40,62,25,49)(15,43,48,51,26,56,47,64)(32,57,46,59,58,63,55,60);(1,5)(2,11)(3,15)(4,21)(6,28)(7,27)(8,29)(9,32)(10,37)(12,40)(13,25)(14,43)(16,26)(17,48)(18,47)(19,49)(20,51)(22,52)(23,54)(24,53)(30,41)(31,57)(33,55)(34,46)(35,58)(36,59)(38,61)(39,62)(42,60)(44,64)(45,56)(50,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,h^a=hi,c^b=cef,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264572);