Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354774 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60948 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1491 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,30)(19,23)(21,28)(27,32)(29,31)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,49)(41,52)(46,60)(47,55)(48,56)(51,57)(53,58)(54,62)(59,61)(63,64);(1,3,6,14,7,15,5,19)(2,8,10,12,11,23,20,22)(4,13,18,28,16,27,17,29)(9,21,25,26,24,31,30,32)(33,35,37,49,39,48,38,47)(34,40,43,45,44,55,42,56)(36,46,52,61,53,62,51,63)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,58)(43,50)(44,57)(45,59)(47,63)(48,62)(49,61)(55,60)(56,64);(4,16)(9,24)(13,27)(17,18)(21,31)(25,30)(26,32)(28,29)(33,37,39,38)(34,42,44,43)(35,47,48,49)(36,51,53,52)(40,45,55,56)(41,50,57,58)(46,61,62,63)(54,64,60,59);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,16,18)(8,22,23,12)(9,25,24,30)(13,28,27,29)(21,32,31,26)(35,48)(40,55)(45,56)(46,62)(47,49)(54,60)(59,64)(61,63)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262027);