Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394813 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14776 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 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)(4,10)(5,24)(6,14)(7,13)(8,12)(9,18)(11,25)(16,40)(17,45)(19,32)(20,31)(21,50)(22,38)(23,36)(26,39)(27,35)(28,37)(29,48)(30,47)(33,43)(34,52)(41,46)(42,57)(44,55)(49,56)(51,59)(53,58)(54,63)(60,62)(61,64);(1,3,8,20,7,19,6,18)(2,9,14,32,13,31,12,15)(4,16,22,43,23,44,27,48)(5,17,28,49,26,47,25,46)(10,33,35,40,36,29,38,55)(11,30,39,56,37,45,24,41)(21,42,51,61,53,63,52,62)(34,57,58,60,59,54,50,64);(1,4,8,22,7,23,6,27)(2,10,14,35,13,36,12,38)(3,16,20,43,19,44,18,48)(5,21,28,51,26,53,25,52)(9,29,32,55,31,33,15,40)(11,34,39,58,37,59,24,50)(17,42,49,61,47,63,46,62)(30,54,56,64,45,57,41,60);(1,5)(2,11)(3,17)(4,21)(6,25)(7,26)(8,28)(9,30)(10,34)(12,24)(13,37)(14,39)(15,41)(16,42)(18,46)(19,47)(20,49)(22,51)(23,53)(27,52)(29,54)(31,45)(32,56)(33,57)(35,58)(36,59)(38,50)(40,60)(43,61)(44,63)(48,62)(55,64)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306345);