Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421911 |
| 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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,33,17,35,18,34)(4,20,23,10,24,39,30,38)(5,25,27,11,29,41,28,40)(15,36,21,48,43,61,44,31)(16,45,47,32,26,56,46,60)(22,51,53,57,55,37,54,49)(42,59,52,64,58,63,50,62);(1,3,7,19,8,17,6,18)(2,9,14,34,12,35,13,33)(4,21,30,43,24,44,23,15)(5,26,28,46,29,16,27,47)(10,36,39,31,38,61,20,48)(11,32,41,45,40,60,25,56)(22,52,54,58,55,50,53,42)(37,62,49,63,51,64,57,59);(1,4)(2,10,12,38)(3,15,17,43)(5,22)(6,23)(7,30)(8,24)(9,31)(11,37,40,51)(13,20,14,39)(16,42,26,58)(18,44,19,21)(25,57,41,49)(27,53)(28,54)(29,55)(32,59)(33,36)(34,61)(35,48)(45,62)(46,50,47,52)(56,64)(60,63);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,32)(10,37)(12,40)(13,41)(14,25)(15,42)(17,26)(18,47)(19,46)(20,49)(21,50)(23,54)(24,55)(30,53)(31,59)(33,45)(34,56)(35,60)(36,62)(38,51)(39,57)(43,58)(44,52)(48,63)(61,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329108);