Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15502 |
| 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 | |
899 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)(4,21,26,42)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25,38,24)(11,31)(16,43,46,32)(17,39)(19,35)(20,34)(22,36,45,60)(23,51,55,49)(28,33)(29,40)(30,41)(37,53,57,54)(44,62,52,58)(47,59)(48,56)(50,61,63,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,39)(12,34)(13,15)(14,35)(16,24)(17,29)(21,32)(23,52)(25,46)(26,45)(27,56)(30,48)(31,47)(33,40)(37,62)(38,60)(41,59)(42,43)(44,55)(49,61)(50,53)(51,64)(54,63)(57,58);(1,4)(2,10,14,38)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32,35,43)(11,37,27,57)(12,21,13,42)(15,36,34,60)(17,44)(18,22)(19,45)(20,46)(28,50)(29,53)(30,54)(31,55)(33,58,59,62)(39,61,56,64)(40,49,41,51)(47,63)(48,52);(1,5,6,29,8,31,7,30)(2,11,12,40,14,27,13,41)(3,17,18,28,20,48,19,47)(4,23,24,53,26,55,25,54)(9,33,15,39,35,59,34,56)(10,37,21,49,38,57,42,51)(16,44,22,50,46,52,45,63)(32,58,36,61,43,62,60,64)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 341662);