Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C2 | C22×C4×D8 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#131 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12,8,14,29,13)(3,15,18,31,20,43,19,49)(4,10,22,36,23,37,6,27)(5,11,26,39,24,38,52,40)(9,30,33,45,34,28,41,16)(17,42,47,56,46,62,48,64)(21,35,51,60,50,59,25,53)(32,55,58,63,57,54,61,44);(1,3)(2,9)(4,16)(5,17)(6,28)(7,18)(8,20)(10,31)(11,32)(12,33)(13,41)(14,34)(15,27)(19,29)(21,44)(22,30)(23,45)(24,46)(25,54)(26,47)(35,56)(36,43)(37,49)(38,57)(39,58)(40,61)(42,53)(48,52)(50,63)(51,55)(59,64)(60,62);(1,4,7,22,8,23,29,6)(2,10,12,36,14,37,13,27)(3,16,18,30,20,45,19,28)(5,21,26,51,24,50,52,25)(9,31,33,43,34,49,41,15)(11,35,39,60,38,59,40,53)(17,44,47,55,46,63,48,54)(32,56,58,62,57,64,61,42);(1,5)(2,11)(3,17)(4,21)(6,25)(7,26)(8,24)(9,32)(10,35)(12,39)(13,40)(14,38)(15,42)(16,44)(18,47)(19,48)(20,46)(22,51)(23,50)(27,53)(28,54)(29,52)(30,55)(31,56)(33,58)(34,57)(36,60)(37,59)(41,61)(43,62)(45,63)(49,64);(5,24)(11,38)(17,46)(21,50)(25,51)(26,52)(32,57)(35,59)(39,40)(42,62)(44,63)(47,48)(53,60)(54,55)(56,64)(58,61)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=hi,g^2=i,h^2=i,b^a=bf,f^a=fh,h^a=hi,f^b=fh,h^b=hi,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320299, #72320305, #72320313, … |
| maximal normal subgroup of | #72320299, #72320305, #72320313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529823);