Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500256 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61646 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
816 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)(3,15)(4,20)(5,9)(6,28)(7,13)(8,14)(10,24)(11,19)(12,17)(16,47)(18,46)(21,54)(22,57)(23,59)(25,43)(26,34)(27,35)(29,53)(30,40)(31,39)(32,44)(33,58)(36,61)(37,51)(38,64)(41,50)(42,62)(45,49)(48,55)(52,56)(60,63);(1,3,8,19)(2,9,14,35)(4,21,25,16)(5,6,27,17)(7,30,32,18)(10,36,39,33)(11,28,15,12)(13,42,44,34)(20,52,43,57)(22,48,56,23)(24,51,31,49)(26,50,62,29)(37,38,45,63)(40,41,46,53)(47,59,54,55)(58,64,61,60);(1,4,6,23,8,25,17,48)(2,10,12,38,14,39,28,63)(3,16,5,22,19,21,27,56)(7,31,29,64,32,24,50,60)(9,33,15,45,35,36,11,37)(13,43,41,59,44,20,53,55)(18,51,26,61,30,49,62,58)(34,52,46,54,42,57,40,47);(1,5,8,27)(2,11,14,15)(3,17,19,6)(4,22,25,56)(7,26,32,62)(9,12,35,28)(10,37,39,45)(13,40,44,46)(16,48,21,23)(18,50,30,29)(20,54,43,47)(24,58,31,61)(33,38,36,63)(34,41,42,53)(49,64,51,60)(52,59,57,55)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407457);