Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468845 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16632 |
| 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 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,34,20,33,18,9)(4,21,23,10,24,41,30,37)(5,25,28,40,29,11,27,39)(16,31,45,42,46,35,51,57)(17,43,49,32,50,59,48,55)(22,38,26,36,53,52,54,61)(44,56,47,62,63,60,64,58);(1,3)(2,9,14,34)(4,16)(5,17)(6,18)(7,19)(8,20)(10,35,37,31)(11,32,25,55)(12,15,13,33)(21,42,41,57)(22,44)(23,45)(24,46)(26,47)(27,48)(28,49)(29,50)(30,51)(36,60,61,56)(38,62,52,58)(39,59,40,43)(53,63)(54,64);(1,4)(2,10,14,37)(3,16)(5,26)(6,30)(7,23)(8,24)(9,31,34,35)(11,38,25,52)(12,41,13,21)(15,42,33,57)(17,47)(18,51)(19,45)(20,46)(22,27)(28,53)(29,54)(32,58,55,62)(36,40,61,39)(43,60,59,56)(44,48)(49,63)(50,64);(1,5,8,29)(2,11,14,25)(3,17,20,50)(4,22,24,53)(6,28,7,27)(9,32,34,55)(10,36,37,61)(12,40,13,39)(15,43,33,59)(16,44,46,63)(18,49,19,48)(21,52,41,38)(23,54,30,26)(31,56,35,60)(42,58,57,62)(45,64,51,47)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376042);