Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396615 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14735 |
| 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 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,37)(4,20,23,10)(5,25,27,38)(6,13,7,12)(9,28,32,18)(11,29,36,26)(16,41,44,39)(17,31,46,58)(21,50,52,56)(22,34,24,35)(30,43,48,53)(33,51,49,54)(40,45,55,47)(42,61,63,57)(59,64,60,62);(1,3)(2,9)(4,16,23,44)(5,17)(6,28)(7,18)(8,19)(10,30,20,48)(11,31)(12,37)(13,15)(14,32)(21,42,52,63)(22,53,24,43)(25,55)(26,47)(27,46)(29,45)(33,57,49,61)(34,41,35,39)(36,58)(38,40)(50,59,56,60)(51,64,54,62);(1,4,6,22,8,23,7,24)(2,10,12,34,14,20,13,35)(3,16,18,43,19,44,28,53)(5,21,26,51,27,52,29,54)(9,30,15,39,32,48,37,41)(11,33,25,56,36,49,38,50)(17,42,45,62,46,63,47,64)(31,57,40,59,58,61,55,60);(1,5)(2,11)(3,17)(4,21)(6,26)(7,29)(8,27)(9,31)(10,33)(12,25)(13,38)(14,36)(15,40)(16,42)(18,45)(19,46)(20,49)(22,51)(23,52)(24,54)(28,47)(30,57)(32,58)(34,56)(35,50)(37,55)(39,59)(41,60)(43,62)(44,63)(48,61)(53,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303812);