Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13907 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1229 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)(3,14,19,9)(4,20)(5,25,28,11)(6,29)(7,13)(8,12)(10,23)(15,34,43,57)(16,32)(17,33,18,41)(21,39,47,31)(22,50,52,48)(24,37)(26,38,27,54)(30,36)(35,51,53,55)(40,46)(42,62)(44,61)(45,58)(49,56)(59,63)(60,64);(1,3,7,18,8,19,6,17)(2,9,13,33,12,14,29,41)(4,21,23,43,24,47,30,15)(5,16,27,45,28,46,26,44)(10,34,37,39,36,57,20,31)(11,32,38,58,25,40,54,61)(22,49,51,63,52,64,55,42)(35,59,50,60,53,62,48,56);(1,4,6,23,8,24,7,30)(2,10,13,20,12,36,29,37)(3,15,17,21,19,43,18,47)(5,22,26,51,28,52,27,55)(9,31,33,57,14,39,41,34)(11,35,38,48,25,53,54,50)(16,42,44,49,46,63,45,64)(32,56,58,62,40,60,61,59);(1,5)(2,11)(3,16)(4,22)(6,26)(7,27)(8,28)(9,32)(10,35)(12,25)(13,38)(14,40)(15,42)(17,44)(18,45)(19,46)(20,48)(21,49)(23,51)(24,52)(29,54)(30,55)(31,56)(33,58)(34,59)(36,53)(37,50)(39,60)(41,61)(43,63)(47,64)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269290);