Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207866 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,25,30)(20,27,29,23)(26,32,28,31)(33,34)(35,45)(36,42)(37,44)(38,40)(39,43)(41,49)(46,59)(47,53)(48,56)(50,54)(51,58)(52,57)(55,61)(60,64)(62,63);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,32)(24,30)(26,29)(27,31)(34,40)(36,50)(41,53)(43,44)(45,56)(46,60)(51,52)(61,62);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(33,35,37,47,38,48,39,49)(34,41,43,56,40,53,44,45)(36,46,52,62,50,60,51,61)(42,55,58,64,54,63,57,59);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,23,24,27)(19,26,25,28)(22,31,30,32)(33,36,38,50)(34,42,40,54)(35,46,48,60)(37,52,39,51)(41,55,53,63)(43,58,44,57)(45,59,56,64)(47,62,49,61);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(33,37,38,39)(34,43,40,44)(35,47,48,49)(36,51,50,52)(41,56,53,45)(42,57,54,58)(46,61,60,62)(55,59,63,64)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fg,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7137405);