Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,18,45)(4,19,21,34)(5,23,25,42)(6,11,28,41)(8,29,33,15)(9,20,35,56)(10,37,39,27)(14,46,49,31)(16,43,17,38)(22,55,57,40)(24,30,51,32)(26,54,58,36)(44,47,53,64)(48,62,52,60)(50,63,59,61);(1,3,7,18)(2,8,12,33)(4,14,21,49)(5,24,25,51)(6,17,28,16)(9,31,35,46)(10,38,39,43)(11,30,41,32)(13,42,45,23)(15,37,29,27)(19,53,34,44)(20,50,56,59)(22,48,57,52)(26,47,58,64)(36,60,54,62)(40,61,55,63);(5,25)(8,30)(10,39)(13,43)(15,29)(19,34)(20,56)(23,42)(24,51)(26,58)(27,37)(31,60)(32,33)(38,45)(40,55)(44,63)(46,62)(47,64)(50,59)(53,61);(1,4,6,22)(2,9,11,36)(3,14,17,48)(5,26,27,20)(7,21,28,57)(8,31,30,60)(10,40,23,34)(12,35,41,54)(13,44,38,61)(15,50,24,47)(16,52,18,49)(19,39,55,42)(25,58,37,56)(29,59,51,64)(32,62,33,46)(43,63,45,53)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153583);