Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425094 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15328 |
| 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 | |
996 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)(3,15,18,33,20,9,19,34)(4,21,26,37,24,40,25,10)(5,27,28,11,30,41,29,39)(16,43,22,31,45,35,46,42)(17,47,48,38,50,57,49,32)(23,36,56,51,54,61,55,58)(44,62,52,63,64,59,53,60);(1,3,8,20)(2,9,12,15)(4,22,24,46)(5,17,30,50)(6,19,7,18)(10,35,37,43)(11,38,39,32)(13,34,14,33)(16,25,45,26)(21,31,40,42)(23,53,54,52)(27,57,41,47)(28,49,29,48)(36,60,61,63)(44,56,64,55)(51,59,58,62);(1,4,6,26,8,24,7,25)(2,10,14,21,12,37,13,40)(3,16,18,22,20,45,19,46)(5,23,28,56,30,54,29,55)(9,31,34,35,15,42,33,43)(11,36,41,51,39,61,27,58)(17,44,48,52,50,64,49,53)(32,59,47,60,38,62,57,63);(1,5)(2,11)(3,17)(4,23)(6,28)(7,29)(8,30)(9,32)(10,36)(12,39)(13,27)(14,41)(15,38)(16,44)(18,48)(19,49)(20,50)(21,51)(22,52)(24,54)(25,55)(26,56)(31,59)(33,57)(34,47)(35,60)(37,61)(40,58)(42,62)(43,63)(45,64)(46,53)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,f^2=i,g^2=i,h^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,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, 332291);