Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,12)(3,15,18,9)(4,21,25,57)(5,26,28,36)(7,31,30,48)(8,33,27,14)(10,38,41,23)(11,19,42,29)(13,44,43,17)(16,50,51,39)(20,37,53,49)(22,47,58,35)(24,59,34,45)(32,46,62,40)(52,63,55,60)(54,56,61,64);(1,3,6,18)(2,9,12,15)(4,16,25,51)(5,17,28,44)(7,29,30,19)(8,20,27,53)(10,35,41,47)(11,31,42,48)(13,26,43,36)(14,37,33,49)(21,50,57,39)(22,23,58,38)(24,52,34,55)(32,61,62,54)(40,56,46,64)(45,60,59,63);(1,4)(2,10)(3,16)(5,22)(6,25)(7,32)(8,34)(9,35)(11,39)(12,41)(13,45)(14,46)(15,47)(17,23)(18,51)(19,54)(20,55)(21,31)(24,27)(26,60)(28,58)(29,61)(30,62)(33,40)(36,63)(37,64)(38,44)(42,50)(43,59)(48,57)(49,56)(52,53);(1,5,27,19,6,28,8,29)(2,11,33,36,12,42,14,26)(3,17,53,7,18,44,20,30)(4,22,24,54,25,58,34,61)(9,31,49,13,15,48,37,43)(10,39,40,63,41,50,46,60)(16,23,52,32,51,38,55,62)(21,56,45,47,57,64,59,35);(3,18)(5,28)(9,15)(11,42)(16,51)(19,29)(20,53)(22,58)(26,36)(35,47)(37,49)(39,50)(52,55)(54,61)(56,64)(60,63)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530350);