Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421913 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15299 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,28,8,12,7,13)(3,9,30,33,16,35,17,34)(4,18,21,10,23,47,22,38)(5,24,26,39,27,11,29,40)(14,41,49,58,43,36,19,31)(15,32,45,59,25,54,44,60)(20,46,51,55,52,62,53,37)(42,61,48,64,56,63,50,57);(1,3,7,30,8,16,6,17)(2,9,28,34,12,35,13,33)(4,19,22,14,23,49,21,43)(5,25,29,44,27,15,26,45)(10,36,47,58,38,41,18,31)(11,32,40,60,24,54,39,59)(20,50,53,42,52,48,51,56)(37,61,46,57,55,63,62,64);(1,4)(2,10,12,38)(3,14,16,43)(5,20)(6,21)(7,22)(8,23)(9,31)(11,37,24,55)(13,18,28,47)(15,42,25,56)(17,19,30,49)(26,51)(27,52)(29,53)(32,57)(33,41)(34,36)(35,58)(39,62,40,46)(44,50,45,48)(54,64)(59,61)(60,63);(1,5)(2,11)(3,15)(4,20)(6,29)(7,26)(8,27)(9,32)(10,37)(12,24)(13,40)(14,42)(16,25)(17,45)(18,46)(19,48)(21,53)(22,51)(23,52)(28,39)(30,44)(31,57)(33,60)(34,59)(35,54)(36,61)(38,55)(41,63)(43,56)(47,62)(49,50)(58,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329110);