Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16316 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,19,42)(4,10,24,38)(5,25,28,11)(6,14,8,12)(9,30,33,18)(16,43,46,34)(17,47,50,44)(20,52,51,31)(21,35,54,61)(22,37,23,36)(26,56,53,39)(27,40,29,41)(32,49,55,58)(45,63,57,60)(48,62,64,59);(1,3,8,30,7,19,6,18)(2,9,14,42,13,33,12,15)(4,20,22,46,24,51,23,16)(5,17,29,58,28,50,27,49)(10,34,36,52,38,43,37,31)(11,32,41,47,25,55,40,44)(21,48,26,57,54,64,53,45)(35,60,39,62,61,63,56,59);(1,4,8,22,7,24,6,23)(2,10,14,36,13,38,12,37)(3,16,30,20,19,46,18,51)(5,26,29,54,28,53,27,21)(9,31,42,34,33,52,15,43)(11,39,41,61,25,56,40,35)(17,48,58,57,50,64,49,45)(32,60,47,62,55,63,44,59);(1,5,6,27,7,28,8,29)(2,11,12,40,13,25,14,41)(3,17,18,49,19,50,30,58)(4,21,22,26,24,54,23,53)(9,32,15,44,33,55,42,47)(10,35,36,39,38,61,37,56)(16,45,20,48,46,57,51,64)(31,59,34,60,52,62,43,63)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365603);