Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472701 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16832 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
706 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)(3,15,19,34)(4,10)(5,24)(6,12)(7,13)(8,14)(9,18,33,29)(11,26)(16,44,47,35)(17,48,51,32)(20,53,46,31)(21,54)(22,37)(23,39)(25,36)(27,42)(28,41)(30,38)(40,55)(43,52,57,50)(45,58,63,62)(49,59,64,60)(56,61);(1,3)(2,9)(4,20,23,46)(5,17)(6,29)(7,18)(8,19)(10,35,39,44)(11,32)(12,34)(13,15)(14,33)(16,22,47,30)(21,49,55,64)(24,57)(25,45,56,63)(26,52)(27,50)(28,51)(31,38,53,37)(36,60,61,59)(40,62,54,58)(41,43)(42,48);(1,4,6,22,8,23,7,30)(2,10,12,37,14,39,13,38)(3,16,29,20,19,47,18,46)(5,25,26,55,28,56,27,21)(9,31,34,44,33,53,15,35)(11,40,41,61,42,54,24,36)(17,49,52,45,51,64,50,63)(32,59,43,58,48,60,57,62);(1,5,8,28)(2,11,14,42)(3,17,19,51)(4,21,23,55)(6,27,7,26)(9,32,33,48)(10,36,39,61)(12,24,13,41)(15,43,34,57)(16,45,47,63)(18,52,29,50)(20,49,46,64)(22,56,30,25)(31,58,53,62)(35,60,44,59)(37,54,38,40)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379898);