Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6379185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,14,18,9,17,33,27,32)(4,19,21,35,22,10,29,36)(5,11,24,40,25,39,28,38)(15,43,47,30,45,41,46,50)(16,42,49,31,48,37,23,56)(20,51,54,60,55,34,53,59)(44,58,52,57,62,61,63,64);(1,3)(2,9)(4,15)(5,23)(6,27)(7,18)(8,17)(10,30)(11,37)(12,33)(13,32)(14,26)(16,24)(19,50)(20,52)(21,47)(22,45)(25,49)(28,48)(29,46)(31,38)(34,58)(35,43)(36,41)(39,42)(40,56)(44,53)(51,61)(54,62)(55,63)(57,59)(60,64);(1,4,8,22)(2,10,13,19)(3,15,17,45)(5,20,25,55)(6,21,7,29)(9,30,32,50)(11,34,39,51)(12,35,26,36)(14,41,33,43)(16,44,48,62)(18,46,27,47)(23,52,49,63)(24,53,28,54)(31,57,56,64)(37,58,42,61)(38,59,40,60);(1,5)(2,11)(3,16)(4,20)(6,28)(7,24)(8,25)(9,31)(10,34)(12,40)(13,39)(14,42)(15,44)(17,48)(18,49)(19,51)(21,54)(22,55)(23,27)(26,38)(29,53)(30,57)(32,56)(33,37)(35,60)(36,59)(41,61)(43,58)(45,62)(46,63)(47,52)(50,64);(3,17)(9,32)(14,33)(15,45)(16,48)(18,27)(23,49)(30,50)(31,56)(37,42)(41,43)(44,62)(46,47)(52,63)(57,64)(58,61)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,h^c=hi,f^d=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72341326, #72341328, #72341363, … |
| maximal normal subgroup of | #72341326, #72341328, #72341363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286388);