Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1525 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)(13,27)(15,23)(16,22)(17,24)(18,25)(19,26)(21,29)(28,31)(30,32)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,58)(45,54)(46,55)(47,61)(50,56)(52,57)(53,60)(59,64)(62,63);(1,3,6,14,7,16,5,15)(2,8,10,12,11,23,20,22)(4,13,17,29,19,30,18,28)(9,21,25,27,26,31,24,32)(33,35)(34,40)(36,44)(37,46)(38,47)(39,45)(41,53)(42,55)(43,54)(48,58)(49,60)(50,59)(51,61)(52,62)(56,64)(57,63);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,21)(10,25)(11,26)(12,27)(14,29)(15,28)(16,30)(20,24)(22,32)(23,31)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,59)(46,60)(47,62)(48,51)(54,63)(55,64)(58,61);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,19,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)(35,45)(40,54)(44,59)(46,47)(53,63)(55,61)(58,64)(60,62);(1,5,7,6)(2,10,11,20)(3,15,16,14)(4,18,19,17)(8,12,23,22)(9,25,26,24)(13,28,30,29)(21,27,31,32)(33,37,39,38)(34,42,43,51)(35,46,45,47)(36,49,50,52)(40,55,54,61)(41,56,57,48)(44,60,59,62)(53,64,63,58)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261900);