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 |
| prime radical | #62864 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,26)(3,14,27,53)(4,10,28,35)(5,22,29,39)(6,11,7,13)(9,30,52,16)(12,24,51,25)(15,41,54,63)(17,40,18,38)(19,34,20,37)(21,48,50,61)(23,32,45,33)(31,55,57,44)(36,47,59,46)(42,60,43,62)(49,56,64,58);(1,3,6,18)(2,9,11,32)(4,15,19,42)(5,23,25,16)(7,17,8,27)(10,31,34,56)(12,38,22,14)(13,33,26,52)(20,43,28,54)(21,49,46,44)(24,30,29,45)(35,57,37,58)(36,60,48,41)(39,53,51,40)(47,55,50,64)(59,62,61,63);(4,19)(5,24)(10,34)(12,39)(14,40)(15,42)(16,45)(20,28)(21,50)(22,51)(23,30)(25,29)(31,56)(35,37)(36,59)(38,53)(41,63)(43,54)(44,55)(46,47)(48,61)(49,64)(57,58)(60,62);(1,4,6,19)(2,10,11,34)(3,15,18,42)(5,21,25,46)(7,20,8,28)(9,31,32,56)(12,36,22,48)(13,37,26,35)(14,41,38,60)(16,44,23,49)(17,43,27,54)(24,47,29,50)(30,55,45,64)(33,58,52,57)(39,61,51,59)(40,62,53,63);(2,11)(4,20)(9,32)(10,35)(12,22)(13,26)(14,38)(15,43)(19,28)(21,47)(31,57)(33,52)(34,37)(36,59)(39,51)(40,53)(41,63)(42,54)(44,64)(46,50)(48,61)(49,55)(56,58)(60,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^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, 6259754);