Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1561 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)(3,13)(4,18,21,9)(5,22)(6,25)(7,12)(8,16)(10,23)(11,26)(14,34,39,56)(15,40)(17,36)(19,46,48,50)(20,32,49,45)(24,51)(27,53,30,38)(28,42)(29,43)(31,47,44,62)(33,52)(35,41)(37,55,60,54)(57,64,63,59)(58,61);(1,3,6,16,7,17,26,43)(2,8,11,13,12,29,25,36)(4,14,20,38,21,39,49,53)(5,15,23,41,24,42,52,61)(9,30,32,56,18,27,45,34)(10,28,22,35,33,40,51,58)(19,37,47,59,48,60,62,64)(31,55,50,63,44,54,46,57);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,27)(10,31)(11,32)(12,18)(13,34)(15,37)(16,38)(17,39)(22,50)(23,47)(24,48)(25,45)(26,49)(28,54)(29,30)(33,44)(35,57)(36,56)(40,55)(41,59)(42,60)(43,53)(46,51)(52,62)(58,63)(61,64);(1,5,6,23,7,24,26,52)(2,10,11,22,12,33,25,51)(3,15,16,41,17,42,43,61)(4,19,20,47,21,48,49,62)(8,28,13,35,29,40,36,58)(9,31,32,50,18,44,45,46)(14,37,38,59,39,60,53,64)(27,54,34,57,30,55,56,63)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267396);