Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,33,18,34)(4,21,23,37,24,40,29,10)(5,25,27,41,28,11,30,39)(16,35,44,58,45,31,48,42)(17,46,26,32,47,38,49,57)(22,50,53,36,54,61,52,55)(43,59,63,62,64,56,51,60);(1,3,8,20)(2,9,14,34)(4,16,24,45)(5,26,28,49)(6,19,7,18)(10,35,37,31)(11,38,25,46)(12,33,13,15)(17,30,47,27)(21,42,40,58)(22,51,54,63)(23,48,29,44)(32,41,57,39)(36,60,55,62)(43,53,64,52)(50,59,61,56);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,31)(11,36)(12,21)(13,40)(14,37)(15,42)(17,43)(18,44)(19,48)(20,45)(25,55)(26,51)(27,52)(28,54)(30,53)(32,56)(33,58)(34,35)(38,62)(39,50)(41,61)(46,60)(47,64)(49,63)(57,59);(1,5,6,27,8,28,7,30)(2,11,12,39,14,25,13,41)(3,17,18,26,20,47,19,49)(4,22,23,52,24,54,29,53)(9,32,15,38,34,57,33,46)(10,36,21,50,37,55,40,61)(16,43,44,51,45,64,48,63)(31,56,42,62,35,59,58,60)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,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, 327004);