Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16768 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,9,18,31,17,32,27,39)(4,19,28,10,22,40,21,35)(5,23,26,37,25,38,24,11)(15,41,45,29,43,56,44,33)(16,46,53,30,48,60,49,57)(20,36,51,59,52,34,54,50)(42,55,62,61,47,58,63,64);(1,3)(2,9,12,32)(4,15,22,43)(5,16)(6,27)(7,18)(8,17)(10,33)(11,30,37,57)(13,39,14,31)(19,41)(20,42,52,47)(21,45,28,44)(23,46,38,60)(24,53)(25,48)(26,49)(29,35)(34,58)(36,55)(40,56)(50,61)(51,63,54,62)(59,64);(1,4)(2,10,12,35)(3,15,17,43)(5,20)(6,21)(7,28)(8,22)(9,29)(11,36,37,34)(13,19,14,40)(16,47,48,42)(18,44,27,45)(23,50,38,59)(24,51)(25,52)(26,54)(30,55)(31,41)(32,33)(39,56)(46,61)(49,63,53,62)(57,58)(60,64);(1,5,6,24,8,25,7,26)(2,11,14,23,12,37,13,38)(3,16,27,53,17,48,18,49)(4,20,21,51,22,52,28,54)(9,30,31,46,32,57,39,60)(10,34,40,59,35,36,19,50)(15,42,45,63,43,47,44,62)(29,55,41,61,33,58,56,64)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=e,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378360);