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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,24)(3,16,20,46)(4,12,32,27)(5,13,33,23)(6,28,34,45)(7,11,8,15)(10,35,39,19)(14,30,44,31)(17,47,52,64)(18,48,54,58)(21,43,22,49)(25,26,59,61)(29,36,56,40)(37,62,55,51)(38,63,50,53)(41,42,60,57);(1,3)(2,10)(4,17)(5,18)(6,29)(7,21)(8,22)(9,20)(11,36)(12,37)(13,38)(14,43)(15,40)(16,28)(19,31)(23,50)(24,39)(25,58)(26,47)(27,55)(30,35)(32,52)(33,54)(34,56)(41,53)(42,62)(44,49)(45,46)(48,59)(51,57)(60,63)(61,64);(2,11)(4,23)(6,30)(10,36)(13,32)(14,44)(15,24)(16,46)(17,50)(19,56)(25,59)(26,60)(28,45)(29,35)(31,34)(38,52)(39,40)(41,61)(42,57)(43,49)(47,63)(48,58)(51,62)(53,64);(1,4)(2,12)(3,17)(5,11)(6,25)(7,23)(8,13)(9,32)(10,37)(14,41)(15,33)(16,47)(18,36)(19,51)(20,52)(21,50)(22,38)(24,27)(26,28)(29,58)(30,42)(31,57)(34,59)(35,62)(39,55)(40,54)(43,53)(44,60)(45,61)(46,64)(48,56)(49,63);(1,5,7,12)(2,13,11,32)(3,18,21,37)(4,24,23,15)(6,26,31,41)(8,27,9,33)(10,38,36,52)(14,42,28,59)(16,48,43,62)(17,39,50,40)(19,53,29,47)(20,54,22,55)(25,44,57,45)(30,60,34,61)(35,63,56,64)(46,58,49,51)Polycyclic presentation:
2^9|a^2=hi,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, 6259641);