Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15602 |
| 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 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,19,35,30,34)(4,20,23,37,24,10,25,41)(5,26,28,40,29,39,31,11)(16,43,46,48,21,42,45,32)(17,33,59,38,47,56,27,58)(22,51,55,36,54,57,53,49)(44,63,64,61,50,60,52,62);(1,3)(2,9)(4,21,24,16)(5,27)(6,30)(7,19)(8,18)(10,32,20,48)(11,38)(12,35)(13,15)(14,34)(17,28)(22,52,54,64)(23,45,25,46)(26,56)(29,59)(31,47)(33,39)(36,61,49,62)(37,42,41,43)(40,58)(44,53,50,55)(51,63,57,60);(1,4,6,25,7,24,8,23)(2,10,12,41,14,20,13,37)(3,16,18,46,19,21,30,45)(5,22,28,55,29,54,31,53)(9,32,15,42,34,48,35,43)(11,36,39,51,40,49,26,57)(17,44,27,52,47,50,59,64)(33,60,38,62,56,63,58,61);(1,5,6,28,7,29,8,31)(2,11,12,39,14,40,13,26)(3,17,18,27,19,47,30,59)(4,22,25,55,24,54,23,53)(9,33,15,38,34,56,35,58)(10,36,41,51,20,49,37,57)(16,44,46,52,21,50,45,64)(32,60,42,62,48,63,43,61)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344748);