Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1627 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)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30)(16,43)(17,33)(19,35)(20,34)(22,50)(23,37)(24,39)(26,38)(28,56)(29,42)(31,41)(32,46)(36,45)(40,47)(44,63)(48,55)(49,52)(51,58)(53,61)(54,62)(57,60)(59,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,25)(17,29)(21,43)(23,44)(24,46)(26,45)(27,55)(30,48)(31,47)(32,38)(33,41)(37,60)(39,50)(42,56)(49,63)(51,52)(53,57)(54,64)(58,62)(59,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,38)(13,21)(14,39)(15,36)(17,44)(18,45)(19,22)(20,46)(27,49)(28,57)(29,52)(30,53)(31,54)(33,58)(34,50)(35,43)(40,59)(41,61)(42,62)(47,51)(48,64)(55,60)(56,63);(1,5,6,29,8,31,7,30)(2,11,12,41,14,42,13,27)(3,17,18,28,20,48,19,47)(4,23,24,52,26,54,25,53)(9,33,15,40,35,55,34,56)(10,37,38,61,39,62,21,49)(16,44,45,57,46,64,22,51)(32,58,36,59,43,60,50,63)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325706);