Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23764 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
428 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)(3,16)(4,22)(5,12)(6,31)(7,14)(8,13)(9,15)(10,19)(11,25)(17,48)(18,37)(20,36)(21,35)(23,58)(24,41)(26,57)(27,42)(28,44)(29,46)(30,43)(32,61)(33,38)(34,54)(39,55)(40,60)(45,53)(47,51)(49,64)(50,62)(52,63)(56,59);(1,3,6,20,9,19,8,33)(2,10,13,36,15,16,31,38)(4,23,25,52,28,59,27,55)(5,18,29,53,30,21,7,32)(11,39,22,56,42,63,44,58)(12,35,14,45,43,37,46,61)(17,49,51,26,50,24,54,60)(34,40,62,41,47,57,48,64);(1,4,9,28)(2,11,15,42)(3,17,19,50)(5,26,30,60)(6,27,8,25)(7,24,29,49)(10,34,16,47)(12,41,43,64)(13,44,31,22)(14,40,46,57)(18,52,21,55)(20,54,33,51)(23,32,59,53)(35,58,37,56)(36,48,38,62)(39,45,63,61);(1,5,8,7,9,30,6,29)(2,12,13,14,15,43,31,46)(3,18,33,32,19,21,20,53)(4,24,27,26,28,49,25,60)(10,35,36,45,16,37,38,61)(11,40,22,41,42,57,44,64)(17,23,54,55,50,59,51,52)(34,39,62,56,47,63,48,58)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=dh,f^c=fi,g^c=ghi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72320520, #72320582, #72320647 |
| maximal normal subgroup of | #72320520, #72320582, #72320647 |
| maximal quotient of | #72321333, #72321505, #72322145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406800);