Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13869 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2047 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)(3,15,19,9)(4,20,24,40)(5,25,28,11)(6,13)(7,14)(8,12)(10,23,36,30)(16,34)(17,32)(18,39,29,33)(21,41)(22,50)(26,38,27,37)(31,44)(35,51)(42,47)(43,62,64,59)(45,57)(46,58)(48,52)(49,56,63,61)(53,55)(54,60);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29)(8,19)(10,34)(11,32)(12,15)(13,33)(14,39)(16,23)(20,31)(22,49)(24,44)(25,42)(26,46)(27,54)(28,47)(30,45)(35,59)(36,57)(37,58)(38,60)(40,41)(43,51)(48,56)(50,61)(52,63)(53,62)(55,64);(1,4)(2,10,12,36)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31,15,41)(11,35,25,53)(13,20,14,40)(17,43)(18,44)(19,45)(21,29)(26,51)(27,55)(28,52)(32,56,42,61)(33,57,39,34)(37,48,38,50)(46,63)(47,64)(49,54)(58,62,60,59);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,32)(10,35)(12,25)(13,37)(14,38)(15,42)(16,43)(18,46)(19,47)(20,48)(21,49)(23,51)(24,52)(29,54)(30,55)(31,56)(33,58)(34,59)(36,53)(39,60)(40,50)(41,61)(44,63)(45,64)(57,62)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267097);