Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,53,37,42,38,43)(35,44,54,56,46,40,47,60)(36,48,52,41,50,59,51,58)(45,61,63,57,49,64,62,55);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,37,46)(34,40,42,44)(36,49,50,45)(38,47,39,54)(41,57,58,55)(43,56,53,60)(48,61,59,64)(51,63,52,62);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(18,23)(19,25)(20,27)(28,32)(33,36,37,50)(34,41,42,58)(35,45,46,49)(38,52,39,51)(40,55,44,57)(43,59,53,48)(47,63,54,62)(56,61,60,64);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,28,29,32)(34,42)(35,46)(41,58)(43,53)(45,49)(47,54)(48,59)(62,63);(1,6,7,5)(2,9,10,4)(3,13,14,15)(8,12,11,16)(19,25)(22,29)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=f,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855182);