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 | #62965 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,36)(3,17,37,62)(4,23,38,5)(6,14,35,58)(7,32,31,47)(8,15,9,16)(11,39,61,20)(12,13,29,26)(18,48,63,54)(19,52,57,60)(21,42,22,50)(24,30,25,27)(28,46,45,34)(33,43,56,44)(40,49,55,51)(41,64,53,59);(1,3)(2,11)(4,18)(5,19)(6,20)(7,33)(8,22)(9,21)(10,37)(12,40)(13,41)(14,42)(15,44)(16,43)(17,32)(23,57)(24,51)(25,49)(26,53)(27,48)(28,59)(29,55)(30,54)(31,56)(34,60)(35,39)(36,61)(38,63)(45,64)(46,52)(47,62)(50,58);(1,4,9,13)(2,12,16,5)(3,18,21,41)(6,24,7,34)(8,26,10,38)(11,40,43,19)(14,45,32,27)(15,23,36,29)(17,48,42,64)(20,51,33,60)(22,53,37,63)(25,31,46,35)(28,47,30,58)(39,49,56,52)(44,57,61,55)(50,59,62,54);(1,5,9,12)(2,13,16,4)(3,19,21,40)(6,30,7,28)(8,29,10,23)(11,41,43,18)(14,46,32,25)(15,38,36,26)(17,49,42,52)(20,54,33,59)(22,55,37,57)(24,58,34,47)(27,31,45,35)(39,48,56,64)(44,63,61,53)(50,60,62,51);(1,6,9,7)(2,14,16,32)(3,20,21,33)(4,24,13,34)(5,27,12,45)(8,31,10,35)(11,42,43,17)(15,47,36,58)(18,51,41,60)(19,48,40,64)(22,56,37,39)(23,30,29,28)(25,26,46,38)(44,62,61,50)(49,53,52,63)(54,55,59,57)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,e^2=h,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, 6260460);