Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187186 |
| 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 | #62743 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,17,26,16,8,19,23)(12,27,29,32,28,21,30,31)(33,34)(35,45)(36,49,52,41)(37,43)(38,42)(39,44)(40,47)(46,60,62,59)(48,58)(50,56,51,57)(53,55)(54,61,63,64);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,27)(16,28)(17,30)(19,29)(22,24)(23,32)(26,31)(33,35)(34,40)(36,46)(37,47)(38,53)(39,48)(41,54)(42,45)(43,58)(44,55)(49,63)(50,61)(51,64)(52,62)(56,59)(57,60);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,15)(10,26)(11,27)(13,28)(14,30)(17,20)(18,29)(22,32)(23,24)(25,31)(33,36)(34,41)(35,46)(37,50)(38,51)(39,52)(40,54)(42,56)(43,57)(44,49)(45,59)(47,61)(48,62)(53,64)(55,63)(58,60);(33,37,39,38)(34,42,44,43)(35,47,48,53)(36,50,52,51)(40,45,55,58)(41,56,49,57)(46,61,62,64)(54,59,63,60)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94565);