Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500349 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24089 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,19,39)(4,20,22,58)(5,26,30,12)(6,31,29,11)(7,32,28,13)(10,40,42,23)(15,48,49,57)(16,37,27,62)(17,36,54,45)(18,46,33,38)(21,60,52,56)(24,44,61,47)(25,43,34,59)(35,53,63,50)(41,51,64,55);(1,3)(2,9)(4,21,22,52)(5,17)(6,27)(7,33)(8,19)(10,41,42,64)(11,37)(12,45)(13,46)(14,39)(15,24,49,61)(16,29)(18,28)(20,56,58,60)(23,51,40,55)(25,50,34,53)(26,36)(30,54)(31,62)(32,38)(35,43,63,59)(44,48,47,57);(1,4,7,34,8,22,28,25)(2,10,13,47,14,42,32,44)(3,15,18,55,19,49,33,51)(5,24,29,40,30,61,6,23)(9,35,38,60,39,63,46,56)(11,43,26,58,31,59,12,20)(16,52,17,53,27,21,54,50)(36,64,37,57,45,41,62,48);(1,3)(2,9)(4,15)(5,27)(6,17)(7,18)(8,19)(10,35)(11,45)(12,37)(13,38)(14,39)(16,30)(20,57)(21,24)(22,49)(23,53)(25,51)(26,62)(28,33)(29,54)(31,36)(32,46)(34,55)(40,50)(41,43)(42,63)(44,56)(47,60)(48,58)(52,61)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249766, #72249781, … |
| maximal normal subgroup of | #72249754, #72249766, #72249781, … |
| maximal quotient of | #72240833, #72249883, #72250027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407546);