Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1964 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1179 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)(3,14,20,39)(4,21)(5,11)(6,27,15,12)(7,30)(8,13)(9,18)(10,24)(16,41,45,62)(17,38,19,48)(22,35)(23,53,40,50)(25,51)(26,56,29,36)(28,33)(31,59)(32,46)(34,55,57,52)(37,42,61,44)(43,49)(47,58)(54,64)(60,63);(1,3,15,18,8,20,6,28)(2,9,27,39,13,33,12,14)(4,16,40,43,25,45,23,54)(5,17,29,46,7,19,26,47)(10,31,57,61,35,60,34,37)(11,32,56,48,30,58,36,38)(21,49,53,62,51,64,50,41)(22,42,55,63,24,44,52,59);(1,4,7,24,8,25,5,22)(2,10,11,21,13,35,30,51)(3,16,19,44,20,45,17,42)(6,23,29,55,15,40,26,52)(9,31,32,49,33,60,58,64)(12,34,36,50,27,57,56,53)(14,37,38,41,39,61,48,62)(18,43,47,59,28,54,46,63);(1,5,8,7)(2,11,13,30)(3,17,20,19)(4,22,25,24)(6,26,15,29)(9,32,33,58)(10,21,35,51)(12,36,27,56)(14,38,39,48)(16,42,45,44)(18,46,28,47)(23,52,40,55)(31,49,60,64)(34,50,57,53)(37,41,61,62)(43,63,54,59)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280935, #72280996, #72281046, … |
| maximal normal subgroup of | #72280935, #72280996, #72281046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402392);