Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | #62867 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,35)(3,18,36,40)(4,12,5,13)(6,14,7,33)(8,16,9,17)(11,37,61,22)(15,31,48,32)(19,54,20,52)(21,41,34,42)(23,53,24,49)(25,43,29,46)(26,47,30,50)(27,44,28,45)(38,58,39,55)(51,57,62,56)(59,64,60,63);(1,3)(2,11)(4,19)(5,20)(6,21)(7,34)(8,24)(9,23)(10,36)(12,38)(13,39)(14,40)(15,49)(16,42)(17,41)(18,33)(22,32)(25,55)(26,59)(27,57)(28,56)(29,58)(30,60)(31,37)(35,61)(43,62)(44,64)(45,63)(46,51)(47,54)(48,53)(50,52);(1,4)(2,12)(3,19)(5,10)(6,25)(7,29)(8,27)(9,28)(11,38)(13,35)(14,43)(15,50)(16,44)(17,45)(18,51)(20,36)(21,55)(22,60)(23,56)(24,57)(26,31)(30,32)(33,46)(34,58)(37,59)(39,61)(40,62)(41,63)(42,64)(47,48)(49,52)(53,54);(1,5,8,28)(2,13,16,45)(3,20,24,56)(4,9,27,10)(6,26,31,25)(7,30,32,29)(11,39,42,63)(12,17,44,35)(14,47,48,43)(15,46,33,50)(18,52,49,51)(19,23,57,36)(21,59,37,55)(22,58,34,60)(38,41,64,61)(40,54,53,62);(1,6,8,31)(2,14,16,48)(3,21,24,37)(4,25,27,26)(5,29,28,30)(7,9,32,10)(11,40,42,53)(12,43,44,47)(13,46,45,50)(15,35,33,17)(18,41,49,61)(19,55,57,59)(20,58,56,60)(22,36,34,23)(38,62,64,54)(39,51,63,52)Polycyclic presentation:
2^9|a^2=hi,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259643);