Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449885 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15952 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,38)(3,13,17,39,16,8,26,31)(4,18,22,48,21,9,32,35)(5,23,24,37,25,40,27,10)(14,29,42,58,19,33,50,60)(15,43,45,30,44,55,46,36)(20,51,53,34,52,56,54,47)(41,61,62,59,49,64,63,57);(1,3)(2,8)(4,19,21,14)(5,15)(6,26)(7,16)(9,33,18,29)(10,36)(11,13)(12,39)(17,28)(20,49,52,41)(22,42,32,50)(23,55)(24,46)(25,44)(27,45)(30,37)(31,38)(34,57,47,59)(35,58,48,60)(40,43)(51,61,56,64)(53,62,54,63);(1,4)(2,9,11,18)(3,14)(5,20)(6,22)(7,21)(8,29,13,33)(10,34,37,47)(12,35,38,48)(15,41)(16,19)(17,42)(23,56,40,51)(24,53)(25,52)(26,50)(27,54)(28,32)(30,57,36,59)(31,58,39,60)(43,64,55,61)(44,49)(45,62)(46,63);(1,5,7,25)(2,10,11,37)(3,15,16,44)(4,20,21,52)(6,27,28,24)(8,30,13,36)(9,34,18,47)(12,40,38,23)(14,41,19,49)(17,46,26,45)(22,54,32,53)(29,57,33,59)(31,43,39,55)(35,51,48,56)(42,63,50,62)(58,64,60,61)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357082);