Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #438234 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15616 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 823 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,35,20,34,19,9)(4,21,24,10,26,38,25,39)(5,27,29,11,30,41,31,42)(16,43,22,51,46,32,45,36)(17,47,28,58,49,33,48,40)(23,53,55,50,57,37,56,62)(44,63,52,59,54,60,64,61);(1,3,6,19,8,20,7,18)(2,9,12,34,14,35,13,15)(4,22,25,46,26,45,24,16)(5,28,29,17,30,48,31,49)(10,36,21,43,39,51,38,32)(11,40,41,33,42,58,27,47)(23,54,56,64,57,44,55,52)(37,61,50,63,53,59,62,60);(1,4,6,24,8,26,7,25)(2,10,13,21,14,39,12,38)(3,16,19,45,20,46,18,22)(5,23,29,55,30,57,31,56)(9,32,15,36,35,43,34,51)(11,37,27,50,42,53,41,62)(17,44,48,64,49,54,28,52)(33,59,40,60,47,61,58,63);(1,5,6,29,8,30,7,31)(2,11,12,41,14,42,13,27)(3,17,18,28,20,49,19,48)(4,23,24,55,26,57,25,56)(9,33,15,40,35,47,34,58)(10,37,38,62,39,53,21,50)(16,44,22,52,46,54,45,64)(32,59,36,60,43,61,51,63)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345430);