Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431933 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#394 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,40)(4,20,25,35)(5,11,26,37)(6,13,7,12)(9,29,32,18)(10,23,36,24)(16,33,21,30)(17,42,45,60)(22,48,51,59)(27,39,28,38)(31,54,56,46)(34,52,58,53)(41,50,47,44)(43,57,49,55)(61,64,63,62);(1,3)(2,9)(4,21)(5,17)(6,29)(7,18)(8,19)(10,33)(11,31)(12,40)(13,15)(14,32)(16,25)(20,47)(22,49)(23,44)(24,50)(26,45)(27,54)(28,46)(30,36)(34,57)(35,41)(37,56)(38,60)(39,42)(43,51)(48,63)(52,62)(53,64)(55,58)(59,61);(1,4,6,23,8,25,7,24)(2,10,12,35,14,36,13,20)(3,16,18,44,19,21,29,50)(5,22,27,52,26,51,28,53)(9,30,15,41,32,33,40,47)(11,34,38,59,37,58,39,48)(17,43,46,62,45,49,54,64)(31,55,42,61,56,57,60,63);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,26)(9,31)(10,34)(12,38)(13,39)(14,37)(15,42)(16,43)(18,46)(19,45)(20,48)(21,49)(23,52)(24,53)(25,51)(29,54)(30,55)(32,56)(33,57)(35,59)(36,58)(40,60)(41,61)(44,62)(47,63)(50,64);(5,26)(11,37)(17,45)(22,51)(27,28)(31,56)(34,58)(38,39)(42,60)(43,49)(46,54)(48,59)(52,53)(55,57)(61,63)(62,64)Polycyclic presentation:
2^9|a^2=i,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,f^b=fi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72307021, #72307032, #72307052, … |
| maximal normal subgroup of | #72307021, #72307032, #72307052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339130);