Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31064 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,29,7,14,5,12,6,13,25,44,26,43,27,11)(3,15,17,9,19,42,30,35,20,46,51,34,28,41,18,33)(4,21,31,58,24,52,22,38,23,10,36,56,53,40,54,39)(16,45,47,32,49,62,57,60,50,61,64,37,55,63,48,59);(1,3)(2,9)(4,16)(5,17)(6,20)(7,28)(8,30)(10,37)(11,41)(12,42)(13,34)(14,33)(15,44)(18,25)(19,26)(21,32)(22,47)(23,50)(24,55)(27,51)(29,46)(31,57)(35,43)(36,48)(38,62)(39,63)(40,60)(45,56)(49,53)(52,59)(54,64)(58,61);(1,4,6,23)(2,10,13,21)(3,16,20,50)(5,22,27,54)(7,24,26,53)(8,31,25,36)(9,32,34,37)(11,38,12,39)(14,40,43,52)(15,45,46,61)(17,47,51,64)(18,48,30,57)(19,49,28,55)(29,56,44,58)(33,59,35,60)(41,63,42,62);(2,11)(3,17)(5,25)(7,26)(8,27)(9,33)(10,38)(12,13)(14,44)(16,47)(18,19)(20,51)(21,39)(22,36)(24,53)(28,30)(29,43)(31,54)(32,59)(34,35)(37,60)(40,58)(41,42)(48,49)(50,64)(52,56)(55,57)(62,63)Polycyclic presentation:
2^9|a^2=efh,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,g^a=gi,c^b=cg,d^b=de,e^b=eh,f^b=fi,h^b=hi,e^d=eh,h^d=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72320520, #72320582, #72320724 |
| maximal normal subgroup of | #72320520, #72320582, #72320724 |
| maximal quotient of | #72321321, #72321498, #72321715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417055);