Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631461 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2840 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1875 conjugacy classes of subgroups; only the 488 normal subgroups are shown.
This group has 1875 conjugacy classes of subgroups — too many to draw. Only its 488 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13,17,8)(4,9)(5,22,25,10)(6,11)(7,12)(14,28,18,38)(15,29)(16,36,26,30)(19,31,23,34)(20,32)(21,33)(24,48,51,35)(27,37)(39,44)(40,52)(41,53,45,57)(42,54)(43,55)(46,56,49,59)(47,58)(50,60)(61,63)(62,64);(1,3,7,17)(2,8,12,13)(4,18,21,14)(5,15,25,44)(6,26,27,16)(9,28,33,38)(10,29,22,39)(11,36,37,30)(19,42,23,40)(20,41,47,45)(24,50,51,43)(31,52,34,54)(32,57,58,53)(35,60,48,55)(46,61,49,62)(56,64,59,63);(1,4)(2,9)(3,14,17,18)(5,23,25,19)(6,20)(7,21)(8,28,13,38)(10,34,22,31)(11,32)(12,33)(15,42)(16,41,26,45)(24,49,51,46)(27,47)(29,54)(30,53,36,57)(35,59,48,56)(37,58)(39,52)(40,44)(43,62)(50,61)(55,64)(60,63);(1,5)(2,10)(3,15)(4,19)(6,24)(7,25)(8,29)(9,31)(11,35)(12,22)(13,39)(14,40)(16,43)(17,44)(18,42)(20,46)(21,23)(26,50)(27,51)(28,52)(30,55)(32,56)(33,34)(36,60)(37,48)(38,54)(41,61)(45,62)(47,49)(53,63)(57,64)(58,59);(1,6,7,27)(2,11,12,37)(3,16,17,26)(4,20,21,47)(5,24,25,51)(8,30,13,36)(9,32,33,58)(10,35,22,48)(14,41,18,45)(15,43,44,50)(19,46,23,49)(28,53,38,57)(29,55,39,60)(31,56,34,59)(40,61,42,62)(52,63,54,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ch,e^b=ei,g^b=gi,d^c=dfh,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538659);