Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,20)(3,12,15,26,16,8,14,23)(4,17,18,24,19,9,21,25)(13,22,28,32,29,27,30,31)(33,34)(35,45)(36,50,52,41)(37,43)(38,42)(39,44)(40,48)(46,59,61,64)(47,56)(49,55)(51,57,53,58)(54,60,63,62);(1,3,7,16)(2,8,11,12)(4,13,19,29)(5,15,6,14)(9,22,17,27)(10,26,20,23)(18,30,21,28)(24,32,25,31)(33,35,39,49)(34,40,44,56)(36,46,52,61)(37,48,38,47)(41,54,50,63)(42,45,43,55)(51,62,53,60)(57,59,58,64);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,27)(14,28)(15,30)(16,29)(20,25)(23,31)(26,32)(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,57)(43,58)(44,50)(45,59)(47,60)(48,62)(49,61)(55,64)(56,63);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,23,12,26)(9,24,17,25)(13,28,29,30)(22,31,27,32)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,55,56,45)(41,57,50,58)(46,60,61,62)(54,64,63,59)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94328);