Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14719 |
| 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 | |
1542 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)(3,14)(4,19)(5,11)(6,27)(7,12)(8,13)(9,17)(10,23)(15,41)(16,40)(18,32)(20,30)(21,35)(22,36)(24,51)(25,39)(26,38)(28,33)(29,37)(31,45)(34,44)(42,58)(43,55)(46,57)(47,50)(48,64)(49,60)(52,56)(53,59)(54,62)(61,63);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,27,32)(4,20,22,15,23,43,29,44)(5,16,24,45,25,46,26,52)(10,34,36,55,19,41,37,30)(11,31,38,40,39,56,51,57)(21,48,49,42,50,62,53,63)(35,58,59,64,47,61,60,54);(1,4,6,22,8,23,7,29)(2,10,12,36,13,19,27,37)(3,15,17,43,18,44,28,20)(5,21,24,49,25,50,26,53)(9,30,14,34,33,55,32,41)(11,35,38,59,39,47,51,60)(16,42,45,62,46,63,52,48)(31,54,40,58,56,64,57,61);(1,5)(2,11)(3,16)(4,21)(6,24)(7,26)(8,25)(9,31)(10,35)(12,38)(13,39)(14,40)(15,42)(17,45)(18,46)(19,47)(20,48)(22,49)(23,50)(27,51)(28,52)(29,53)(30,54)(32,57)(33,56)(34,58)(36,59)(37,60)(41,61)(43,62)(44,63)(55,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301999);