Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461427 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16391 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,29)(3,14,18,34,17,9,28,33)(4,10,22,37,21,38,23,54)(5,24,27,11,26,40,30,39)(15,31,19,35,44,41,51,60)(16,45,47,32,49,56,48,42)(20,52,53,36,25,57,55,62)(43,61,50,63,46,64,58,59);(1,3)(2,9)(4,19)(5,16)(6,28)(7,18)(8,17)(10,35)(11,32)(12,14)(13,34)(15,22)(20,50)(21,51)(23,44)(24,56)(25,58)(26,49)(27,48)(29,33)(30,47)(31,37)(36,61)(38,60)(39,42)(40,45)(41,54)(43,53)(46,55)(52,63)(57,59)(62,64);(1,4)(2,10)(3,15,17,44)(5,25,26,20)(6,22)(7,23)(8,21)(9,31,14,41)(11,36,39,62)(12,38)(13,37)(16,46)(18,19,28,51)(24,52,40,57)(27,55,30,53)(29,54)(32,59)(33,35,34,60)(42,63)(43,49)(45,64)(47,58)(48,50)(56,61);(1,5,8,26)(2,11,12,39)(3,16,17,49)(4,20,21,25)(6,27,7,30)(9,32,14,42)(10,36,38,62)(13,40,29,24)(15,43,44,46)(18,47,28,48)(19,50,51,58)(22,53,23,55)(31,59,41,63)(33,56,34,45)(35,61,60,64)(37,57,54,52)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,e^c=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, 368624);