Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351869 |
| 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 | C42 | C25 |
| prime radical | #62828 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2610 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)(3,13)(4,17)(5,12)(6,11)(7,10)(8,16)(9,19)(14,22)(15,25)(18,26)(20,23)(21,24)(27,30)(28,32)(29,31)(33,34,39,43)(35,45,49,54)(36,41,51,57)(37,44,38,42)(40,47,55,48)(46,60,62,59)(50,58,52,56)(53,63,64,61);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,23)(11,25)(12,13)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35)(34,40)(36,46)(37,48)(38,47)(39,49)(41,53)(42,45)(43,55)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,24)(11,17)(12,26)(13,27)(15,28)(16,29)(20,30)(23,32)(25,31)(33,36)(34,41)(35,46)(37,50)(38,52)(39,51)(40,53)(42,56)(43,57)(44,58)(45,59)(47,61)(48,63)(49,62)(54,60)(55,64);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(33,37,39,38)(34,42,43,44)(35,47,49,48)(36,50,51,52)(40,54,55,45)(41,56,57,58)(46,61,62,63)(53,60,64,59);(2,10)(4,18)(8,23)(11,12)(13,25)(14,28)(19,21)(29,30)(34,43)(36,51)(40,55)(42,44)(45,54)(46,62)(50,52)(61,63)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259080);