Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201948 |
| 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 | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
966 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,19)(14,28)(16,24)(17,23)(20,25)(21,26)(22,30)(27,29)(31,32)(33,34,38,43,39,42,37,44)(35,45,48,55,49,58,47,40)(36,50,52,41,53,57,51,56)(46,54,61,64,62,63,60,59);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,19,29,20,30,21,31)(9,22,18,27,26,28,25,32)(33,35)(34,40,42,55)(36,46)(37,47)(38,48)(39,49)(41,54,56,63)(43,45,44,58)(50,59,57,64)(51,60)(52,61)(53,62);(1,4,7,20)(2,9,12,26)(3,14,17,30)(5,19,6,21)(8,22,24,28)(10,25,11,18)(13,27,23,32)(15,29,16,31)(33,36,39,53)(34,41,42,56)(35,46,49,62)(37,52,38,51)(40,54,55,63)(43,50,44,57)(45,59,58,64)(47,61,48,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,25,26,18)(14,29,30,31)(22,32,28,27)(34,42)(40,55)(41,56)(43,44)(45,58)(50,57)(54,63)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109300);