Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13869 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,19,33,20,9,18,34)(4,21,24,37,26,40,25,10)(5,27,28,39,29,11,30,38)(16,43,45,31,46,35,22,41)(17,32,47,58,48,42,49,59)(23,52,54,56,55,50,53,36)(44,60,51,57,64,62,63,61);(1,3,8,20)(2,9,12,15)(4,22,26,45)(5,17,29,48)(6,19,7,18)(10,35,37,43)(11,32,27,42)(13,34,14,33)(16,24,46,25)(21,31,40,41)(23,51,55,63)(28,49,30,47)(36,60,56,62)(38,59,39,58)(44,53,64,54)(50,57,52,61);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31,15,41)(11,36,27,56)(13,21,14,40)(17,44)(18,45)(19,22)(20,46)(28,53)(29,55)(30,54)(32,57,42,61)(33,35,34,43)(38,50,39,52)(47,63)(48,64)(49,51)(58,60,59,62);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,38)(14,39)(15,42)(16,44)(18,47)(19,49)(20,48)(21,50)(22,51)(24,53)(25,54)(26,55)(31,57)(33,58)(34,59)(35,60)(37,56)(40,52)(41,61)(43,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267102);