Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16012 |
| 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 | |
865 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)(3,14,26,9,18,39,17,32)(4,19,21,46,22,10,28,35)(5,11,24,37,25,38,29,53)(15,40,42,33,43,57,50,30)(16,36,23,51,44,61,52,31)(20,45,54,59,49,34,48,63)(41,60,64,55,62,58,47,56);(1,3)(2,9)(4,15)(5,23)(6,26)(7,18)(8,17)(10,33)(11,36)(12,39)(13,32)(14,27)(16,24)(19,30)(20,47)(21,50)(22,43)(25,52)(28,42)(29,44)(31,37)(34,58)(35,40)(38,61)(41,48)(45,60)(46,57)(49,64)(51,53)(54,62)(55,63)(56,59);(1,4,6,21,7,22,8,28)(2,10,12,35,13,19,27,46)(3,15,17,42,18,43,26,50)(5,20,24,48,25,49,29,54)(9,30,14,40,32,33,39,57)(11,34,37,59,38,45,53,63)(16,41,23,47,44,62,52,64)(31,55,36,60,51,56,61,58);(1,5,7,25)(2,11)(3,16)(4,20,22,49)(6,24,8,29)(9,31,32,51)(10,34)(12,37)(13,38)(14,36,39,61)(15,41)(17,23)(18,44)(19,45)(21,48,28,54)(26,52)(27,53)(30,55,33,56)(35,59)(40,60,57,58)(42,47)(43,62)(46,63)(50,64)Polycyclic presentation:
2^9|a^2=egh,c^2=e,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 358950);