Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2369 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)(4,10,23,37)(5,24)(6,27)(7,12)(8,13)(9,17)(11,25)(15,41,44,33)(16,31)(18,32)(19,40)(20,30,48,55)(21,50,52,60)(22,36,29,35)(26,39)(28,38)(34,53,59,51)(42,45)(43,54,63,64)(46,57)(47,56)(49,58,62,61);(1,3,8,19)(2,9,13,32)(4,20,23,48)(5,16,26,46)(6,18,7,17)(10,33,37,41)(11,31,38,57)(12,40,27,14)(15,29,44,22)(21,49,52,62)(24,42,39,56)(25,47,28,45)(30,35,55,36)(34,58,59,61)(43,51,63,53)(50,54,60,64);(1,4)(2,10,13,37)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30,32,55)(11,34,38,59)(12,35,27,36)(14,41,40,33)(16,43)(17,20)(18,48)(19,44)(24,50,39,60)(25,51)(26,52)(28,53)(31,54,57,64)(42,61,56,58)(45,62)(46,63)(47,49);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,26)(9,31)(10,34)(12,39)(13,38)(14,42)(15,43)(17,47)(18,45)(19,46)(20,49)(22,53)(23,52)(24,27)(29,51)(30,54)(32,57)(33,58)(35,60)(36,50)(37,59)(40,56)(41,61)(44,63)(48,62)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281215);