Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1853 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1853 |
| 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 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,7,13,26,12)(3,14,18,42,19,43,16,39)(4,20,21,37,24,36,23,9)(6,27,25,11,29,38,49,40)(8,30,33,17,34,28,32,50)(15,44,47,41,48,62,45,63)(22,35,53,52,56,61,55,57)(31,46,59,54,60,64,51,58);(1,3,7,19)(2,8)(4,15,24,48)(5,16,26,18)(6,28,29,30)(9,31)(10,32)(11,39)(12,33)(13,34)(14,27)(17,49,50,25)(20,51)(21,45,23,47)(22,54,56,58)(35,44)(36,59)(37,60)(38,43)(40,42)(41,57)(46,55,64,53)(52,63)(61,62);(1,4)(2,9)(3,15)(5,21)(6,22)(7,24)(8,31)(10,20)(11,35)(12,36)(13,37)(14,41)(16,45)(17,46)(18,47)(19,48)(23,26)(25,53)(27,57)(28,54)(29,56)(30,58)(32,51)(33,59)(34,60)(38,52)(39,44)(40,61)(42,62)(43,63)(49,55)(50,64);(1,5,7,26)(2,10,13,12)(3,16,19,18)(4,21,24,23)(6,25,29,49)(8,32,34,33)(9,20,37,36)(11,38,40,27)(14,39,43,42)(15,45,48,47)(17,30,50,28)(22,53,56,55)(31,51,60,59)(35,52,61,57)(41,44,63,62)(46,58,64,54)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gi,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280256, #72280389, #72281021, … |
| maximal normal subgroup of | #72280256, #72280389, #72281021, … |
| maximal quotient of | C2×#494277, #72310095, #72321794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401473);