Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2826 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1314 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)(3,15)(4,21)(5,27)(6,12)(7,13)(8,14)(9,19)(10,25)(11,29)(16,45)(17,48)(18,42)(20,34)(22,43)(23,54)(24,38)(26,37)(28,39)(30,40)(31,41)(32,53)(33,52)(35,47)(36,56)(44,50)(46,61)(49,59)(51,57)(55,60)(58,63)(62,64);(1,3,6,18,8,19,7,20)(2,9,13,42,14,15,12,34)(4,22,25,16,26,53,24,47)(5,17,29,50,30,51,31,52)(10,35,21,32,38,45,37,43)(11,33,27,57,41,44,40,48)(23,49,28,46,56,64,55,63)(36,59,39,58,54,62,60,61);(1,4)(2,10)(3,16)(5,28)(6,24)(7,25)(8,26)(9,32)(11,39)(12,21)(13,37)(14,38)(15,43)(17,49)(18,22)(19,47)(20,53)(23,29)(27,36)(30,55)(31,56)(33,59)(34,45)(35,42)(40,54)(41,60)(44,62)(46,52)(48,58)(50,63)(51,64)(57,61);(1,5)(2,11)(3,17)(4,23)(6,29)(7,31)(8,30)(9,33)(10,36)(12,40)(13,27)(14,41)(15,44)(16,46)(18,50)(19,51)(20,52)(21,39)(22,49)(24,55)(25,28)(26,56)(32,58)(34,48)(35,59)(37,60)(38,54)(42,57)(43,61)(45,62)(47,63)(53,64);(1,6,8,7)(2,12,14,13)(3,18,19,20)(4,24,26,25)(5,29,30,31)(9,34,15,42)(10,21,38,37)(11,40,41,27)(16,22,47,53)(17,50,51,52)(23,55,56,28)(32,45,43,35)(33,48,44,57)(36,39,54,60)(46,49,63,64)(58,62,61,59)Polycyclic presentation:
2^9|b^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,d^c=dfh,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538651);