Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467840 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16592 |
| 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 | |
791 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)(3,15)(4,21,26,38)(5,27,30,41)(6,13)(7,12)(8,14)(9,18)(10,24,39,25)(11,29,42,31)(16,32,46,52)(17,33,50,60)(19,35)(20,34)(22,36,51,43)(23,37)(28,56)(40,54)(44,49,59,48)(45,58)(47,57)(53,62)(55,61)(63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,25,16,26,51,24,46)(5,17,31,49,30,50,29,48)(10,36,21,52,39,43,38,32)(11,33,27,44,42,60,41,59)(23,53,28,45,55,47,54,64)(37,58,40,62,61,63,56,57);(1,4,6,24,8,26,7,25)(2,10,12,38,14,39,13,21)(3,16,18,22,20,46,19,51)(5,28,29,55,30,54,31,23)(9,32,15,43,35,52,34,36)(11,40,41,61,42,56,27,37)(17,47,48,64,50,53,49,45)(33,58,59,62,60,63,44,57);(1,5)(2,11,14,42)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33,35,60)(10,37,39,61)(12,27,13,41)(15,44,34,59)(16,45)(18,49)(19,48)(20,50)(21,40,38,56)(22,53)(24,54)(25,28)(26,55)(32,57,52,62)(36,58,43,63)(46,64)(47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=efg,c^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375037);