Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14725 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,27,37)(3,8,15,30,16,31,46,58)(4,17,21,33,20,35,25,9)(5,22,23,36,24,10,26,34)(13,38,40,55,18,48,44,28)(14,41,42,57,43,29,45,56)(19,47,51,32,50,59,52,53)(39,60,61,54,49,64,62,63);(1,3)(2,8)(4,18,20,13)(5,14)(6,15)(7,16)(9,28,33,55)(10,29)(11,30)(12,31)(17,38,35,48)(19,49,50,39)(21,44,25,40)(22,41)(23,42)(24,43)(26,45)(27,46)(32,54,53,63)(34,56)(36,57)(37,58)(47,60,59,64)(51,62,52,61);(1,4)(2,9)(3,13)(5,19)(6,25)(7,20)(8,28)(10,32)(11,35)(12,33)(14,39)(15,44)(16,18)(17,37)(21,27)(22,53)(23,52)(24,50)(26,51)(29,54)(30,48)(31,55)(34,47)(36,59)(38,58)(40,46)(41,63)(42,62)(43,49)(45,61)(56,60)(57,64);(1,5,7,24)(2,10,12,22)(3,14,16,43)(4,19,20,50)(6,26,27,23)(8,29,31,41)(9,32,33,53)(11,36,37,34)(13,39,18,49)(15,45,46,42)(17,47,35,59)(21,52,25,51)(28,54,55,63)(30,57,58,56)(38,60,48,64)(40,62,44,61)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302727);