Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498916 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2067 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)(3,14)(4,19)(5,25)(6,30)(7,13)(8,11)(9,16)(10,22)(12,29)(15,47)(17,52)(18,40)(20,45)(21,56)(23,38)(24,44)(26,34)(27,54)(28,42)(31,46)(32,43)(33,55)(35,51)(36,50)(37,60)(39,59)(41,58)(48,62)(49,63)(53,64)(57,61);(1,3)(2,9)(4,20)(5,26)(6,17)(7,31)(8,18)(10,37)(11,34)(12,40)(13,35)(14,25)(15,23)(16,29)(19,33)(21,57)(22,49)(24,53)(27,55)(28,51)(30,36)(32,50)(38,62)(39,47)(41,63)(42,52)(43,46)(44,61)(45,56)(48,58)(54,64)(59,60);(1,4,8,24)(2,10,11,38)(3,15,18,49)(5,27,29,21)(6,23,7,22)(9,33,34,61)(12,41,25,39)(13,44,30,19)(14,45,40,64)(16,48,26,60)(17,53,31,20)(28,58,32,59)(35,37,36,62)(42,54,43,56)(46,63,52,47)(50,57,51,55);(2,11)(9,34)(10,38)(12,25)(13,30)(14,40)(19,44)(33,61)(35,36)(37,62)(39,41)(42,43)(45,64)(46,52)(47,63)(54,56)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=eh,f^c=fi,g^c=gi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72303720, #72303728, #72303737, … |
| maximal normal subgroup of | #72303720, #72303728, #72303737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406113);