Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16839 |
| 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 | |
785 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)(3,15)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,24)(11,29)(16,32)(17,33)(19,35)(20,34)(22,36)(23,55)(25,38)(26,42)(28,39)(30,41)(31,40)(37,57)(43,46)(44,49)(45,63)(47,52)(48,64)(50,62)(51,61)(53,56)(54,59)(58,60);(1,3,7,18,8,20,6,19)(2,9,12,34,14,35,13,15)(4,22,24,16,26,46,25,47)(5,17,29,51,31,50,30,49)(10,36,42,32,38,43,21,52)(11,33,40,61,41,62,27,44)(23,54,56,45,57,48,28,58)(37,60,39,59,55,63,53,64);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,42)(3,16,18,22,20,47,19,46)(5,28,29,57,31,56,30,23)(9,32,15,43,35,52,34,36)(11,39,27,55,41,53,40,37)(17,48,49,58,50,54,51,45)(33,60,61,64,62,63,44,59);(1,5,6,30,8,31,7,29)(2,11,12,40,14,41,13,27)(3,17,19,49,20,50,18,51)(4,23,24,56,26,57,25,28)(9,33,34,61,35,62,15,44)(10,37,21,53,38,55,42,39)(16,45,46,48,47,58,22,54)(32,59,36,60,52,64,43,63)Polycyclic presentation:
2^9|b^2=egh,c^2=e,d^2=f,e^2=i,f^2=i,g^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,h^b=hi,d^c=de,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380066);