Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,19,32,29,33)(4,20,23,35,24,40,22,10)(5,11,27,37,28,39,26,38)(16,30,44,41,46,59,45,58)(17,42,25,55,47,36,48,31)(21,50,53,61,54,49,52,34)(43,63,51,60,64,57,56,62);(1,3)(2,9)(4,16)(5,25)(6,29)(7,19)(8,18)(10,30)(11,36)(12,32)(13,15)(14,33)(17,26)(20,41)(21,51)(22,45)(23,44)(24,46)(27,47)(28,48)(31,37)(34,60)(35,59)(38,55)(39,42)(40,58)(43,52)(49,63)(50,57)(53,64)(54,56)(61,62);(1,4,8,22,7,24,6,23)(2,10,13,20,14,35,12,40)(3,16,29,44,19,46,18,45)(5,21,26,53,28,54,27,52)(9,30,32,58,33,59,15,41)(11,34,37,49,39,61,38,50)(17,43,25,56,47,64,48,51)(31,57,36,62,55,63,42,60);(1,5)(2,11)(3,17)(4,21)(6,27)(7,28)(8,26)(9,31)(10,34)(12,38)(13,37)(14,39)(15,42)(16,43)(18,48)(19,47)(20,49)(22,53)(23,52)(24,54)(25,29)(30,57)(32,36)(33,55)(35,61)(40,50)(41,60)(44,56)(45,51)(46,64)(58,62)(59,63)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,h^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,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 293107);