Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201121 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
995 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)(3,11)(4,15,16,8)(5,18)(6,10)(7,13)(9,20)(12,25,27,32)(14,26)(17,23,19,24)(21,28,31,29)(22,30)(33,34)(35,39,45,53)(36,40)(37,48)(38,42)(41,50)(43,51,58,63)(44,59,61,52)(46,56)(47,55)(49,54)(57,64,60,62);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,16,27,17,28)(8,21,24,25,15,31,23,32)(33,35)(34,39,42,53)(36,43)(37,44)(38,45)(40,51,55,63)(41,52,48,59)(46,57)(47,58)(49,60)(50,61)(54,62,56,64);(1,4,6,16)(2,8,10,15)(3,12,14,27)(5,19,20,17)(7,21,22,31)(9,24,18,23)(11,25,26,32)(13,29,30,28)(33,36,38,47)(34,40,42,55)(35,43,45,58)(37,49,50,46)(39,51,53,63)(41,56,48,54)(44,60,61,57)(52,64,59,62);(4,16)(8,15)(12,27)(17,19)(21,31)(23,24)(25,32)(28,29)(33,37,38,50)(34,41,42,48)(35,44,45,61)(36,46,47,49)(39,52,53,59)(40,54,55,56)(43,57,58,60)(51,62,63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108323);