Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#2034 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1061 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)(3,15)(4,21)(5,11)(6,27)(7,12)(8,14)(9,17)(10,24)(13,30)(16,46)(18,35)(19,34)(20,36)(22,56)(23,42)(25,55)(26,43)(28,41)(29,38)(31,40)(32,54)(33,60)(37,58)(39,59)(44,48)(45,51)(47,62)(49,64)(50,63)(52,53)(57,61);(1,3,7,20,8,19,5,17)(2,9,11,34,14,36,12,15)(4,22,23,52,25,16,32,47)(6,28,30,49,31,18,26,51)(10,33,38,44,39,63,37,61)(13,41,27,45,43,35,40,64)(21,53,54,56,55,62,42,46)(24,48,58,60,59,57,29,50);(1,4,5,23,8,25,7,32)(2,10,12,38,14,39,11,37)(3,16,17,47,19,22,20,52)(6,29,26,24,31,58,30,59)(9,33,15,44,36,63,34,61)(13,42,40,21,43,54,27,55)(18,50,49,48,28,60,51,57)(35,56,45,62,41,46,64,53);(1,5,8,7)(2,11,14,12)(3,17,19,20)(4,23,25,32)(6,26,31,30)(9,34,36,15)(10,37,39,38)(13,27,43,40)(16,47,22,52)(18,49,28,51)(21,42,55,54)(24,58,59,29)(33,61,63,44)(35,64,41,45)(46,62,56,53)(48,60,57,50)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249772, #72249776, #72249832, … |
| maximal normal subgroup of | #72249772, #72249776, #72249832, … |
| maximal quotient of | #72250208, C2×#499058, #72322979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406254);