Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424321 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,31)(3,14,17,35,18,34,30,9)(4,19,22,38,24,49,23,10)(5,25,27,11,29,40,28,41)(15,32,44,42,20,36,51,60)(16,45,26,57,47,33,46,39)(21,37,53,48,55,63,54,56)(43,64,58,61,50,62,52,59);(1,3,8,18)(2,9,13,35)(4,20,24,15)(5,26,29,46)(6,30,7,17)(10,36,38,32)(11,39,41,57)(12,34,31,14)(16,28,47,27)(19,42,49,60)(21,52,55,58)(22,44,23,51)(25,45,40,33)(37,62,63,64)(43,53,50,54)(48,61,56,59);(1,4)(2,10,13,38)(3,15,18,20)(5,21)(6,22)(7,23)(8,24)(9,32)(11,37,41,63)(12,19,31,49)(14,42)(16,43,47,50)(17,44,30,51)(25,56,40,48)(26,58,46,52)(27,53)(28,54)(29,55)(33,59)(34,60)(35,36)(39,64)(45,61)(57,62);(1,5,6,27,8,29,7,28)(2,11,12,40,13,41,31,25)(3,16,17,26,18,47,30,46)(4,21,22,53,24,55,23,54)(9,33,14,39,35,45,34,57)(10,37,19,48,38,63,49,56)(15,43,44,58,20,50,51,52)(32,59,42,64,36,61,60,62)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331518);