Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14516 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8,15,29)(4,17)(5,10,22,37)(6,12)(7,11)(9,20)(13,27,41,56)(14,28)(16,30,24,38)(18,31,49,45)(19,46,51,33)(21,47)(23,36,25,39)(26,32)(34,53)(35,42)(40,55)(43,57)(44,58)(48,59)(50,64)(52,61,54,60)(62,63);(1,3)(2,8,11,29)(4,18)(5,14,22,42)(6,24)(7,15)(9,31,17,45)(10,35)(12,38,32,30)(13,21)(16,26)(19,50,51,48)(20,49)(23,44,25,43)(27,47,56,34)(28,37)(33,59)(36,57)(39,58)(40,54,63,52)(41,53)(46,64)(55,61)(60,62);(1,4,6,21,7,20,26,53)(2,9,32,47,11,17,12,34)(3,13,16,18,15,41,24,49)(5,19,25,54,22,51,23,52)(8,27,38,45,29,56,30,31)(10,33,36,60,37,46,39,61)(14,40,44,48,42,63,43,50)(28,55,57,64,35,62,58,59);(1,5,6,25,7,22,26,23)(2,10,32,36,11,37,12,39)(3,14,16,44,15,42,24,43)(4,19,21,54,20,51,53,52)(8,28,38,57,29,35,30,58)(9,33,47,60,17,46,34,61)(13,40,18,48,41,63,49,50)(27,55,45,64,56,62,31,59)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=efg,d^2=efg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295325);