Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347398 |
| 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 | #62831 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10)(3,12,16,25)(4,9,17,24)(5,11,6,20)(8,14,22,15)(13,26,27,32)(18,23,19,30)(21,28,31,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,55)(44,51)(45,50)(46,53)(47,52)(48,54)(49,57)(56,63)(58,61)(59,62)(60,64);(1,3,7,16)(2,8,10,22)(4,13,17,27)(5,15,6,14)(9,21,24,31)(11,25,20,12)(18,29,19,28)(23,32,30,26)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,60)(54,61)(59,64)(62,63);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,26)(14,28)(15,29)(16,27)(20,30)(22,31)(25,32)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,55)(51,60)(54,62)(58,64)(61,63);(2,10)(4,17)(8,22)(11,20)(12,25)(13,27)(18,19)(28,29)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(1,5,7,6)(2,11,10,20)(3,14,16,15)(4,18,17,19)(8,12,22,25)(9,23,24,30)(13,28,27,29)(21,26,31,32)(33,37,38,48)(34,42,41,54)(35,44,45,58)(36,46,47,59)(39,51,50,61)(40,53,52,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254620);