Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14130 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,33,20,40,18,9)(4,21,25,10,26,36,24,37)(5,27,29,39,30,38,28,11)(16,31,44,41,45,50,22,34)(17,46,48,60,49,32,47,42)(23,35,54,56,55,59,53,51)(43,61,62,64,63,58,52,57);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,40)(4,22,24,45,26,44,25,16)(5,17,28,47,30,49,29,48)(10,34,36,31,37,41,21,50)(11,32,27,42,39,46,38,60)(23,52,53,63,55,62,54,43)(35,58,56,57,59,61,51,64);(1,4,7,25,8,26,6,24)(2,10,12,36,14,37,13,21)(3,16,19,44,20,45,18,22)(5,23,29,54,30,55,28,53)(9,31,15,41,33,50,40,34)(11,35,27,56,39,59,38,51)(17,43,48,62,49,63,47,52)(32,57,42,61,46,64,60,58);(1,5,6,28,8,30,7,29)(2,11,12,27,14,39,13,38)(3,17,18,47,20,49,19,48)(4,23,24,53,26,55,25,54)(9,32,15,42,33,46,40,60)(10,35,36,56,37,59,21,51)(16,43,22,52,45,63,44,62)(31,57,41,61,50,64,34,58)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281076);