Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198171 |
| 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 | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
968 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)(4,17,20,25)(5,10)(6,11)(7,12)(8,15)(9,19,26,18)(14,28,30,22)(16,23)(21,24)(27,29,32,31)(33,34)(35,39,45,54)(36,40)(37,48)(38,42)(41,51)(43,52,60,63)(44,57,49,53)(46,58)(47,56)(50,55)(59,64,61,62);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,19,31,20,30,18,29)(9,22,25,27,26,28,17,32)(33,35)(34,39,42,54)(36,43)(37,49)(38,45)(40,52,56,63)(41,57,48,53)(44,51)(46,61)(47,60)(50,59)(55,64,58,62);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,24,28)(10,17,11,25)(13,27,23,32)(15,31,21,29)(33,36,38,47)(34,40,42,56)(35,43,45,60)(37,50,51,46)(39,52,54,63)(41,58,48,55)(44,61,49,59)(53,64,57,62);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,25,26,17)(14,29,30,31)(22,27,28,32)(33,37,38,51)(34,41,42,48)(35,44,45,49)(36,46,47,50)(39,53,54,57)(40,55,56,58)(43,59,60,61)(52,62,63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105370);