Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62903 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,27)(3,14,28,54)(4,11,19,46)(5,23,29,38)(6,10,7,13)(9,30,53,16)(12,25,52,26)(15,40,41,61)(17,39,18,37)(20,48,47,58)(21,34,22,35)(24,31,45,33)(32,55,63,42)(36,50,64,51)(43,60,44,59)(49,56,62,57);(1,3,6,18)(2,9,10,31)(4,15,21,44)(5,24,26,16)(7,17,8,28)(11,32,34,56)(12,37,23,14)(13,33,27,53)(19,41,22,43)(20,49,51,42)(25,30,29,45)(35,57,46,63)(36,59,48,40)(38,54,52,39)(47,62,50,55)(58,61,64,60);(5,25)(12,38)(14,39)(16,45)(20,50)(23,52)(24,30)(26,29)(36,58)(37,54)(40,60)(42,62)(47,51)(48,64)(49,55)(59,61);(2,10)(4,19)(9,31)(11,35)(12,23)(13,27)(14,37)(15,41)(20,47)(21,22)(32,57)(33,53)(34,46)(36,58)(38,52)(39,54)(40,60)(42,55)(43,44)(48,64)(49,62)(50,51)(56,63)(59,61);(1,4,7,22)(2,11,13,35)(3,15,17,43)(5,20,25,50)(6,21,8,19)(9,32,33,57)(10,34,27,46)(12,36,38,58)(14,40,39,60)(16,42,45,62)(18,44,28,41)(23,48,52,64)(24,49,30,55)(26,51,29,47)(31,56,53,63)(37,59,54,61)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260451);