Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | C2×#1519 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,8,13,7,30)(3,14,18,38,19,43,16,42)(4,20,23,35,24,10,21,37)(6,26,28,12,29,39,25,40)(9,27,32,50,34,49,33,17)(15,44,45,61,48,41,47,62)(22,54,53,36,57,58,56,52)(31,55,60,46,51,63,59,64);(1,3)(2,9)(4,15)(5,16)(6,27)(7,18)(8,19)(10,31)(11,32)(12,38)(13,34)(14,26)(17,28)(20,51)(21,45)(22,55)(23,47)(24,48)(25,50)(29,49)(30,33)(35,59)(36,61)(37,60)(39,43)(40,42)(41,58)(44,54)(46,56)(52,62)(53,64)(57,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31)(11,35)(12,36)(13,20)(14,41)(16,45)(17,46)(18,47)(19,48)(25,53)(26,58)(27,55)(28,56)(29,57)(30,37)(32,59)(33,60)(34,51)(38,61)(39,54)(40,52)(42,62)(43,44)(49,63)(50,64);(1,5,8,7)(2,11,13,30)(3,16,19,18)(4,21,24,23)(6,25,29,28)(9,32,34,33)(10,35,20,37)(12,26,40,39)(14,42,43,38)(15,45,48,47)(17,27,50,49)(22,53,57,56)(31,59,51,60)(36,58,52,54)(41,62,44,61)(46,55,64,63)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307086, #72307090, #72307091, … |
| maximal normal subgroup of | #72307086, #72307090, #72307091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389234);