Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| 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 | #62926 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2595 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)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34,39,43)(35,45,49,53)(36,42,50,58)(37,44,38,41)(40,48,55,47)(46,59,60,63)(51,57,52,56)(54,62,64,61);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25)(33,35,39,49)(34,40,43,55)(36,46,50,60)(37,48,38,47)(41,53,44,45)(42,54,58,64)(51,62,52,61)(56,63,57,59);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,23)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32)(34,41)(36,50)(37,38)(40,53)(42,57)(43,44)(45,55)(46,60)(47,48)(54,59)(56,58)(63,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,36)(34,42)(35,46)(37,51)(38,52)(39,50)(40,54)(41,56)(43,58)(44,57)(45,59)(47,61)(48,62)(49,60)(53,63)(55,64);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)(34,43)(36,50)(40,55)(41,44)(45,53)(46,60)(51,52)(61,62)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258331);