Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16476 |
| 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 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,27,7,13,6,12)(3,14,25,9,18,39,17,31)(4,10,26,40,21,35,20,34)(5,11,23,37,24,38,28,54)(15,41,49,32,45,56,44,29)(16,42,47,58,48,62,53,30)(19,33,22,36,50,60,52,61)(43,63,46,55,64,57,51,59);(1,3,7,18)(2,9,13,31)(4,15,21,45)(5,16,24,48)(6,25,8,17)(10,32,35,29)(11,30,38,58)(12,39,27,14)(19,43,50,64)(20,49,26,44)(22,51,52,46)(23,53,28,47)(33,59,60,55)(34,56,40,41)(36,57,61,63)(37,62,54,42);(1,4,6,26,7,21,8,20)(2,10,27,34,13,35,12,40)(3,15,17,49,18,45,25,44)(5,22,23,19,24,52,28,50)(9,29,39,56,31,32,14,41)(11,36,54,60,38,61,37,33)(16,46,47,43,48,51,53,64)(30,57,62,59,58,63,42,55);(1,5)(2,11,13,38)(3,16)(4,19,21,50)(6,23)(7,24)(8,28)(9,30,31,58)(10,33)(12,37,27,54)(14,42,39,62)(15,43,45,64)(17,47)(18,48)(20,22,26,52)(25,53)(29,55)(32,59)(34,36)(35,60)(40,61)(41,63)(44,46,49,51)(56,57)Polycyclic presentation:
2^9|a^2=efg,b^2=i,c^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370728);