Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1807 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1807 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
933 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)(3,14,18,39)(4,20,24,33)(5,11)(6,26)(7,29)(8,13)(9,17,32,46)(10,23,35,21)(12,27)(15,40)(16,38,19,48)(22,34,53,60)(25,36)(28,55)(30,57)(31,45,56,47)(37,41)(42,58)(43,62)(44,61)(49,64)(50,51,54,52)(59,63);(1,3,6,17,7,19,27,47,8,18,28,46,5,16,25,45)(2,9,12,14,11,31,26,38,13,32,36,39,29,56,55,48)(4,15,22,42,23,43,52,57,24,44,53,64,21,41,51,63)(10,30,34,37,33,58,54,61,35,59,60,62,20,49,50,40);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,30)(11,33)(12,34)(13,35)(14,37)(16,41)(17,42)(18,44)(19,43)(20,29)(25,51)(26,54)(27,52)(28,53)(31,58)(32,59)(36,60)(38,61)(39,62)(40,48)(45,63)(46,64)(47,57)(49,56)(50,55);(1,5,8,7)(2,11,13,29)(3,16,18,19)(4,21,24,23)(6,25,28,27)(9,31,32,56)(10,33,35,20)(12,26,36,55)(14,38,39,48)(15,41,44,43)(17,45,46,47)(22,51,53,52)(30,58,59,49)(34,54,60,50)(37,61,62,40)(42,63,64,57)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285601, #72285749, #72285824, … |
| maximal normal subgroup of | #72285601, #72285749, #72285824, … |
| maximal quotient of | #72310106, #72321759, #72323039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401003);