Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1845 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)(3,15)(4,10)(5,13)(6,12)(7,28)(8,14)(9,18)(11,24)(16,47)(17,32)(19,45)(20,54)(21,43)(22,37)(23,39)(25,58)(26,41)(27,40)(29,46)(30,38)(31,52)(33,60)(34,53)(35,55)(36,56)(42,51)(44,50)(48,64)(49,61)(57,63)(59,62);(1,3)(2,9)(4,20)(5,17)(6,18)(7,29)(8,19)(10,35)(11,32)(12,15)(13,42)(14,34)(16,21)(22,55)(23,48)(24,51)(25,59)(26,53)(27,60)(28,33)(30,50)(31,38)(36,47)(37,54)(39,62)(40,46)(41,45)(43,61)(44,63)(49,56)(52,57)(58,64);(1,4)(2,10)(3,16)(5,21)(6,25)(7,30)(8,23)(9,31)(11,36)(12,37)(13,43)(14,39)(15,44)(17,48)(18,52)(19,50)(20,29)(22,26)(24,57)(27,56)(28,38)(32,54)(33,62)(34,61)(35,42)(40,63)(41,58)(45,47)(46,64)(49,53)(51,55)(59,60);(1,5,8,7)(2,11,14,40)(3,17,19,29)(4,21,23,30)(6,24,26,27)(9,32,34,46)(10,36,39,63)(12,13,41,28)(15,42,45,33)(16,48,50,20)(18,51,53,60)(22,56,25,57)(31,54,61,64)(35,47,62,44)(37,43,58,38)(49,59,52,55)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281946, #72282072, #72282105, … |
| maximal normal subgroup of | #72281946, #72282072, #72282105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407019);