Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432256 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15453 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 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)(3,8,15,28)(4,9)(5,20)(6,23)(7,11)(10,22)(12,25)(13,36,38,26)(14,40,41,53)(16,44,43,29)(17,46)(18,31)(19,51)(21,35)(24,34)(27,45,33,42)(30,48)(32,50)(37,60,47,64)(39,55,62,59)(49,57)(52,56)(54,61,58,63);(1,3)(2,8,11,28)(4,13,18,38)(5,14,21,41)(6,16)(7,15)(9,26)(10,33)(12,29,23,44)(17,47)(19,39,50,62)(20,53)(22,42,24,45)(25,43)(27,34)(30,54,56,58)(31,36)(32,55)(35,40)(37,49)(46,64,57,60)(48,63)(51,59)(52,61);(1,4)(2,9)(3,13,15,38)(5,17)(6,19)(7,18)(8,26,28,36)(10,30)(11,31)(12,32)(14,37,41,47)(16,39,43,62)(20,46)(21,49)(22,48)(23,51)(24,52)(25,50)(27,54,33,58)(29,55,44,59)(34,56)(35,57)(40,64,53,60)(42,61,45,63);(1,5,7,21)(2,10,11,34)(3,14)(4,17,18,49)(6,24,25,22)(8,27)(9,30,31,56)(12,35,23,20)(13,37)(15,41)(16,45)(19,52,50,48)(26,54)(28,33)(29,53)(32,57,51,46)(36,58)(38,47)(39,63)(40,44)(42,43)(55,60)(59,64)(61,62)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339453);