Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364625 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| 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 | |
2307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,17,9,18,33,19,34)(4,20,23,37,25,10,24,38)(5,26,27,11,29,40,28,39)(15,35,44,57,43,48,21,31)(16,32,47,58,46,59,45,41)(22,51,53,36,54,55,52,49)(42,56,61,60,62,64,50,63);(1,3,7,17,8,18,6,19)(2,9,12,33,13,34,30,14)(4,21,23,15,25,44,24,43)(5,16,27,47,29,46,28,45)(10,35,38,57,20,48,37,31)(11,32,40,58,39,59,26,41)(22,50,53,42,54,61,52,62)(36,60,55,64,49,63,51,56);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,20)(14,35)(16,42)(17,21)(18,43)(19,44)(26,55)(27,52)(28,53)(29,54)(30,38)(32,56)(33,48)(34,57)(39,49)(40,51)(41,60)(45,61)(46,62)(47,50)(58,63)(59,64);(1,5,8,29)(2,11,13,39)(3,16,18,46)(4,22,25,54)(6,28,7,27)(9,32,34,59)(10,36,20,49)(12,40,30,26)(14,41,33,58)(15,42,43,62)(17,47,19,45)(21,50,44,61)(23,53,24,52)(31,56,57,64)(35,60,48,63)(37,51,38,55)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=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, 271824);