Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15273 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,33,19,41,30,9)(4,20,23,37,24,36,22,10)(5,25,28,11,29,39,27,40)(16,31,45,49,46,34,44,42)(17,32,47,58,48,59,26,38)(21,51,53,50,55,35,54,56)(43,62,52,60,63,61,64,57);(1,3)(2,9)(4,16)(5,26)(6,30)(7,18)(8,19)(10,34)(11,38)(12,41)(13,33)(14,15)(17,27)(20,49)(21,52)(22,45)(23,44)(24,46)(25,32)(28,48)(29,47)(31,37)(35,61)(36,42)(39,59)(40,58)(43,53)(50,57)(51,62)(54,63)(55,64)(56,60);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,35)(12,36)(13,37)(14,20)(15,42)(17,43)(18,44)(19,46)(25,56)(26,52)(27,53)(28,54)(29,55)(30,45)(32,57)(33,34)(38,62)(39,50)(40,51)(41,49)(47,64)(48,63)(58,61)(59,60);(1,5,6,27,8,29,7,28)(2,11,12,25,13,40,14,39)(3,17,18,26,19,48,30,47)(4,21,22,53,24,55,23,54)(9,32,15,38,33,59,41,58)(10,35,36,56,37,51,20,50)(16,43,44,52,46,63,45,64)(31,57,42,62,34,60,49,61)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327002);