Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514144 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58990 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3008 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)(3,14)(4,20)(5,25)(6,8)(7,13)(9,18)(10,23)(11,26)(12,29)(15,33)(16,30)(17,46)(19,32)(21,50)(22,36)(24,34)(27,38)(28,37)(31,48)(35,52)(39,41)(40,47)(42,57)(43,55)(44,59)(45,56)(49,51)(53,58)(54,61)(60,63)(62,64);(2,8)(3,15)(6,29)(10,34)(11,25)(12,13)(14,32)(16,43)(17,42)(18,19)(20,36)(22,23)(26,27)(35,58)(37,38)(39,55)(40,57)(41,45)(44,61)(47,48)(49,50)(51,53)(59,64)(60,62);(1,3)(2,9)(4,16)(5,17)(6,15)(7,19)(8,14)(10,30)(11,31)(12,32)(13,33)(18,29)(20,39)(21,44)(22,41)(23,43)(24,45)(25,40)(26,47)(27,42)(28,48)(34,55)(35,54)(36,56)(37,57)(38,46)(49,59)(50,63)(51,61)(52,62)(53,60)(58,64);(1,4,7,24)(2,10,13,36)(3,16,19,45)(5,21,28,52)(6,22,29,23)(8,20,12,34)(9,30,33,56)(11,35,38,50)(14,39,32,55)(15,41,18,43)(17,44,48,62)(25,49,37,58)(26,51,27,53)(31,54,46,63)(40,59,57,64)(42,60,47,61);(1,5)(2,11)(3,17)(4,21)(6,27)(7,28)(8,25)(9,31)(10,35)(12,37)(13,38)(14,40)(15,42)(16,44)(18,47)(19,48)(20,49)(22,53)(23,51)(24,52)(26,29)(30,54)(32,57)(33,46)(34,58)(36,50)(39,59)(41,60)(43,61)(45,62)(55,64)(56,63)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,f^b=fi,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421342);