Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394805 |
| 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 | |
1544 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)(10,22)(15,42)(16,41)(18,32)(20,50)(21,52)(23,36)(24,39)(25,38)(26,40)(28,33)(29,37)(30,45)(31,46)(34,44)(35,54)(43,56)(47,58)(48,57)(49,55)(51,59)(53,60)(61,63)(62,64);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,27,32)(4,20,22,15,23,44,29,45)(5,16,24,46,26,48,25,47)(10,34,19,42,37,50,36,30)(11,31,38,41,40,58,39,57)(21,51,53,43,55,62,54,63)(35,59,49,61,60,64,52,56);(1,4,6,22,8,23,7,29)(2,10,12,19,13,37,27,36)(3,15,17,44,18,45,28,20)(5,21,24,53,26,55,25,54)(9,30,14,34,33,42,32,50)(11,35,38,49,40,60,39,52)(16,43,46,62,48,63,47,51)(31,56,41,59,58,61,57,64);(1,5)(2,11)(3,16)(4,21)(6,24)(7,25)(8,26)(9,31)(10,35)(12,38)(13,40)(14,41)(15,43)(17,46)(18,48)(19,49)(20,51)(22,53)(23,55)(27,39)(28,47)(29,54)(30,56)(32,57)(33,58)(34,59)(36,52)(37,60)(42,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 302001);