Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | #62963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,6,20)(5,17,18,10)(8,21,24,13)(14,28,29,26)(16,30,19,25)(23,27,31,32)(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)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,27)(17,31)(18,29)(19,32)(25,28)(26,30)(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);(1,4,7,6)(2,9,11,20)(3,13,15,21)(5,16,18,19)(8,22,24,12)(10,25,17,30)(14,27,29,32)(23,28,31,26)(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);(1,5)(2,10)(3,14)(4,16)(6,19)(7,18)(8,23)(9,25)(11,17)(12,26)(13,27)(15,29)(20,30)(21,32)(22,28)(24,31)(37,48)(41,54)(44,58)(46,59)(50,61)(52,62)(56,64)(60,63);(5,18)(10,17)(14,29)(16,19)(23,31)(25,30)(26,28)(27,32)(33,37,38,48)(34,41,42,54)(35,44,45,58)(36,46,47,59)(39,50,51,61)(40,52,53,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260461);