Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62865 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34,38,42)(35,39,45,51)(36,40,47,53)(37,41,48,54)(43,55,57,49)(44,50,58,61)(46,52,59,62)(56,63,64,60);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,60)(53,55)(54,61)(59,64)(62,63);(1,4,6,15)(2,8,9,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(10,23,17,29)(11,25,24,32)(13,27,18,30)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,60)(51,55)(54,62)(58,64)(61,63);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,37)(34,41)(35,44)(36,46)(38,48)(39,50)(40,52)(42,54)(43,56)(45,58)(47,59)(49,60)(51,61)(53,62)(55,63)(57,64);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)(37,48)(41,54)(44,58)(46,59)(50,61)(52,62)(56,64)(60,63)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259724);