Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14175 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,6,12)(3,9,19,32,18,33,17,31)(4,20,25,10,23,36,24,35)(5,11,26,37,27,38,28,39)(15,40,43,29,21,48,42,55)(16,30,44,56,45,57,46,58)(22,50,52,59,51,47,53,34)(41,61,62,64,49,60,63,54);(1,3)(2,9)(4,21,23,15)(5,16)(6,17)(7,19)(8,18)(10,29,35,55)(11,30)(12,31)(13,33)(14,32)(20,40,36,48)(22,49,51,41)(24,43,25,42)(26,44)(27,45)(28,46)(34,54,59,64)(37,56)(38,57)(39,58)(47,60,50,61)(52,63,53,62);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,29)(11,34)(12,36)(13,35)(14,20)(16,41)(17,43)(18,21)(19,42)(26,53)(27,51)(28,52)(30,54)(31,48)(32,40)(33,55)(37,47)(38,59)(39,50)(44,63)(45,49)(46,62)(56,60)(57,64)(58,61);(1,5,7,26,8,27,6,28)(2,11,14,37,13,38,12,39)(3,16,19,44,18,45,17,46)(4,22,24,53,23,51,25,52)(9,30,32,56,33,57,31,58)(10,34,20,47,35,59,36,50)(15,41,42,63,21,49,43,62)(29,54,40,60,55,64,48,61)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283375);