Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 | |
2624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14)(3,17,35,62)(4,5,24,11)(6,29,32,45)(8,15,9,16)(10,36,46,20)(12,28,25,27)(13,34,44,33)(18,48,60,63)(19,53,37,51)(21,42,22,50)(23,43,58,61)(26,31,41,59)(30,39,56,40)(38,64,52,47)(49,55,57,54);(1,3,8,21)(2,10,15,39)(4,18,25,52)(5,19,27,54)(6,30,33,20)(7,35,9,22)(11,37,28,55)(12,38,24,60)(13,42,29,17)(14,46,16,40)(23,57,59,51)(26,47,43,48)(31,53,58,49)(32,56,34,36)(41,64,61,63)(44,50,45,62);(1,4,8,25)(2,11,15,28)(3,18,21,52)(5,16,27,14)(6,31,33,58)(7,24,9,12)(10,37,39,55)(13,43,29,26)(17,47,42,48)(19,40,54,46)(20,49,30,53)(22,38,35,60)(23,32,59,34)(36,57,56,51)(41,44,61,45)(50,63,62,64);(1,4)(2,5)(3,18)(6,23)(7,24)(8,25)(9,12)(10,19)(11,14)(13,26)(15,27)(16,28)(17,48)(20,51)(21,52)(22,38)(29,43)(30,57)(31,34)(32,58)(33,59)(35,60)(36,53)(37,46)(39,54)(40,55)(41,44)(42,47)(45,61)(49,56)(50,64)(62,63);(1,5)(2,12)(3,19)(4,16)(6,26)(7,11)(8,27)(9,28)(10,38)(13,31)(14,25)(15,24)(17,49)(18,40)(20,48)(21,54)(22,55)(23,45)(29,58)(30,47)(32,41)(33,43)(34,61)(35,37)(36,63)(39,60)(42,53)(44,59)(46,52)(50,51)(56,64)(57,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259752);