Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14453 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,28)(3,14,19,40,17,9,18,32)(4,10,23,36,24,35,22,53)(5,25,26,38,27,11,29,39)(15,30,44,33,43,41,20,48)(16,37,46,56,45,31,47,57)(21,50,51,60,52,34,54,61)(42,55,49,59,63,62,64,58);(1,3,6,19,8,17,7,18)(2,9,28,32,12,14,13,40)(4,20,22,15,24,44,23,43)(5,16,26,47,27,45,29,46)(10,33,53,41,35,48,36,30)(11,37,39,57,25,31,38,56)(21,49,51,42,52,64,54,63)(34,59,61,55,50,58,60,62);(1,4,6,22,8,24,7,23)(2,10,28,53,12,35,13,36)(3,15,18,20,17,43,19,44)(5,21,26,51,27,52,29,54)(9,30,40,48,14,41,32,33)(11,34,39,61,25,50,38,60)(16,42,46,49,45,63,47,64)(31,55,57,59,37,62,56,58);(1,5)(2,11)(3,16)(4,21)(6,26)(7,29)(8,27)(9,31)(10,34)(12,25)(13,38)(14,37)(15,42)(17,45)(18,46)(19,47)(20,49)(22,51)(23,54)(24,52)(28,39)(30,55)(32,56)(33,58)(35,50)(36,60)(40,57)(41,62)(43,63)(44,64)(48,59)(53,61)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294386);