Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22261 |
| prime radical | #23794 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,15,35,49,9,16,36,50,6,13,33,47)(3,10,19,5,18,29,54,28,20,38,56,30,12,43,45,32)(4,21,26,57,25,41,61,48,24,46,34,62,31,11,39,42)(17,51,52,59,53,37,63,60,22,58,55,23,44,40,64,27);(1,3,7,19,8,18,35,54,9,20,36,56,6,12,33,45)(2,10,14,5,15,29,49,28,16,38,50,30,13,43,47,32)(4,22,34,52,25,44,39,63,24,17,26,55,31,53,61,64)(11,40,48,60,21,51,62,23,41,37,42,27,46,58,57,59);(1,4,6,31,9,24,8,25)(2,11,13,46,16,41,15,21)(3,17,18,53,20,22,12,44)(5,27,28,59,30,60,32,23)(7,34,33,61,36,26,35,39)(10,37,29,58,38,40,43,51)(14,48,47,57,50,42,49,62)(19,55,54,64,56,52,45,63);(1,5)(2,12)(3,15)(4,23,24,59)(6,32)(7,29)(8,28)(9,30)(10,33)(11,22,41,17)(13,20)(14,45)(16,18)(19,49)(21,44,46,53)(25,27,31,60)(26,40,34,37)(35,38)(36,43)(39,58,61,51)(42,55,48,52)(47,56)(50,54)(57,64,62,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ce,f^b=fi,d^c=dh,e^c=ei,f^c=fi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224962 |
| maximal normal subgroup of | #72224962 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404321);