Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199851 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,14,27,15,8)(4,17,20,9,18,25,19,24)(13,22,29,26,30,32,28,31)(33,34)(35,44,47,56)(36,48)(37,43)(38,52)(39,42)(40,46,55,51)(41,49)(45,60,63,54)(50,58)(53,57)(59,61,64,62);(1,3,7,14)(2,8)(4,13,18,30)(5,15,6,16)(9,22)(10,12)(11,23)(17,31)(19,28,20,29)(21,27)(24,32)(25,26)(33,35,37,51,39,47,38,46)(34,40,43,56,42,55,52,44)(36,45,49,62,50,63,53,61)(41,54,58,64,57,60,48,59);(1,4)(2,9)(3,13)(5,19)(6,20)(7,18)(8,22)(10,25)(11,24)(12,26)(14,30)(15,28)(16,29)(17,21)(23,32)(27,31)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,52)(51,62)(55,60)(56,64);(3,14)(4,18)(8,23)(9,24)(12,27)(15,16)(17,25)(19,20)(34,42)(40,55)(41,57)(43,52)(44,56)(48,58)(54,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107111);