Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #414109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15162 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,25,40)(5,26,29,38)(6,13,7,12)(9,19,34,18)(10,30,37,24)(11,27,39,28)(16,31,22,35)(17,46,48,32)(23,52,55,36)(41,44,58,45)(42,49,59,47)(43,62,51,61)(50,54,56,53)(57,63,60,64);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,41)(23,51)(24,44)(26,42)(27,49)(28,47)(29,48)(30,45)(31,37)(36,60)(38,59)(39,46)(40,58)(43,55)(50,61)(52,57)(53,63)(54,64)(56,62);(1,4,6,24,8,25,7,30)(2,10,12,21,14,37,13,40)(3,16,18,44,20,22,19,45)(5,23,27,53,29,55,28,54)(9,31,15,41,34,35,33,58)(11,36,38,50,39,52,26,56)(17,43,47,63,48,51,49,64)(32,57,42,61,46,60,59,62);(1,5,7,28,8,29,6,27)(2,11,13,26,14,39,12,38)(3,17,18,47,20,48,19,49)(4,23,30,54,25,55,24,53)(9,32,15,42,34,46,33,59)(10,36,40,56,37,52,21,50)(16,43,44,63,22,51,45,64)(31,57,41,61,35,60,58,62)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321306);