Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16851 |
| 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 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,18,34,19,43,17,9)(4,20,24,36,25,39,23,10)(5,26,29,40,30,11,28,38)(15,32,46,51,21,53,45,41)(16,42,49,33,50,61,48,57)(22,37,55,62,56,52,27,35)(44,59,58,60,54,63,47,64);(1,3,7,18,8,19,6,17)(2,9,31,43,13,34,12,14)(4,21,24,45,25,15,23,46)(5,16,28,48,30,50,29,49)(10,32,39,41,36,53,20,51)(11,33,38,61,26,57,40,42)(22,54,27,58,56,44,55,47)(35,59,37,60,62,63,52,64);(1,4,8,25)(2,10,13,36)(3,15,19,21)(5,27,30,55)(6,24,7,23)(9,32,34,53)(11,37,26,52)(12,39,31,20)(14,41,43,51)(16,47,50,58)(17,46,18,45)(22,29,56,28)(33,60,57,64)(35,40,62,38)(42,63,61,59)(44,49,54,48);(1,5)(2,11,13,26)(3,16)(4,22)(6,29)(7,28)(8,30)(9,33,34,57)(10,35,36,62)(12,40,31,38)(14,42,43,61)(15,44)(17,49)(18,48)(19,50)(20,52,39,37)(21,54)(23,55)(24,27)(25,56)(32,59,53,63)(41,60,51,64)(45,58)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380355);