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 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,10,13,7)(4,8,5,14)(11,16,12,15)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34)(35,43)(36,41)(37,39)(38,50)(40,47)(42,44)(45,56)(46,55)(48,52)(49,61)(51,57)(53,60)(54,58)(59,63)(62,64);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,31)(24,32)(26,29)(27,30)(34,39)(36,48)(40,51)(42,50)(43,55)(45,59)(49,58)(60,64);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,23)(19,25)(24,27)(26,28)(33,35,44,57,37,46,38,47)(34,40,50,55,39,51,42,43)(36,45,58,64,48,59,49,60)(41,53,61,63,52,62,54,56);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,24,23,27)(19,26,25,28)(22,30,31,32)(33,36,37,48)(34,41,39,52)(35,45,46,59)(38,49,44,58)(40,53,51,62)(42,54,50,61)(43,56,55,63)(47,60,57,64);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,31)(26,28)(30,32)(34,39)(36,48)(40,51)(42,50)(43,55)(45,59)(49,58)(60,64)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fgh,d^2=fh,e^2=fi,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, 7139198);