Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15578 |
| 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 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,25,6,11,24,12)(3,8,15,46,16,31,44,30)(4,17,26,34,20,38,19,9)(5,10,22,39,23,37,27,36)(13,40,47,32,42,49,43,28)(14,35,45,57,21,29,48,58)(18,51,54,60,52,50,53,33)(41,61,63,56,55,64,62,59);(1,3,7,15,6,16,24,44)(2,8,12,30,11,31,25,46)(4,13,19,43,20,42,26,47)(5,21,22,48,23,14,27,45)(9,32,17,49,34,28,38,40)(10,35,36,58,37,29,39,57)(18,41,53,62,52,55,54,63)(33,59,51,61,60,56,50,64);(1,4)(2,9)(3,13)(5,18)(6,20)(7,26)(8,28)(10,33)(11,34)(12,38)(14,41)(15,47)(16,42)(17,25)(19,24)(21,55)(22,54)(23,52)(27,53)(29,56)(30,49)(31,32)(35,59)(36,50)(37,60)(39,51)(40,46)(43,44)(45,63)(48,62)(57,61)(58,64);(1,5,7,27,6,23,24,22)(2,10,12,39,11,37,25,36)(3,14,44,45,16,21,15,48)(4,18,26,53,20,52,19,54)(8,29,46,57,31,35,30,58)(9,33,38,51,34,60,17,50)(13,41,43,63,42,55,47,62)(28,56,40,61,32,59,49,64)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344192);