Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16149 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
687 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)(3,15,19,32,20,40,18,9)(4,21,28,10,24,34,23,35)(5,11,26,38,27,39,29,37)(16,30,44,57,45,58,43,41)(17,46,25,53,48,36,47,31)(22,49,51,61,52,60,55,33)(42,63,50,59,64,56,54,62);(1,3,6,18,7,20,8,19)(2,9,12,15,14,32,13,40)(4,16,23,43,24,45,28,44)(5,25,29,17,27,47,26,48)(10,30,34,57,35,58,21,41)(11,36,38,31,39,46,37,53)(22,50,55,42,52,54,51,64)(33,59,49,56,61,62,60,63);(1,4,7,24)(2,10,14,35)(3,16,20,45)(5,22,27,52)(6,28,8,23)(9,30,32,58)(11,33,39,61)(12,21,13,34)(15,41,40,57)(17,42,48,64)(18,44,19,43)(25,54,47,50)(26,55,29,51)(31,56,53,63)(36,62,46,59)(37,49,38,60);(1,5,7,27)(2,11)(3,17)(4,22,24,52)(6,26,8,29)(9,31,32,53)(10,33)(12,37)(13,38)(14,39)(15,36,40,46)(16,42)(18,25)(19,47)(20,48)(21,49)(23,51,28,55)(30,56,58,63)(34,60)(35,61)(41,62,57,59)(43,50)(44,54)(45,64)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361040);