Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629051 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2847 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4380 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)(3,15,18,32)(4,21,24,37)(5,11,26,36)(6,13,7,12)(9,20,31,19)(10,27,34,23)(16,41,43,29)(17,40,45,57)(22,50,53,49)(25,38,28,35)(30,48,55,46)(33,51,58,52)(39,47,56,44)(42,54,61,60)(59,62,64,63);(1,3,6,19,8,18,7,20)(2,9,12,15,14,31,13,32)(4,16,23,44,24,43,27,47)(5,17,25,46,26,45,28,48)(10,29,21,39,34,41,37,56)(11,30,35,40,36,55,38,57)(22,42,51,62,53,61,52,63)(33,54,49,59,58,60,50,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,27)(8,24)(9,29)(11,33)(12,21)(13,37)(14,34)(15,39)(17,42)(18,43)(19,44)(20,47)(25,51)(26,53)(28,52)(30,54)(31,41)(32,56)(35,49)(36,58)(38,50)(40,59)(45,61)(46,62)(48,63)(55,60)(57,64);(1,5)(2,11)(3,17)(4,22)(6,25)(7,28)(8,26)(9,30)(10,33)(12,35)(13,38)(14,36)(15,40)(16,42)(18,45)(19,46)(20,48)(21,49)(23,51)(24,53)(27,52)(29,54)(31,55)(32,57)(34,58)(37,50)(39,59)(41,60)(43,61)(44,62)(47,63)(56,64);(3,18)(9,31)(15,32)(16,43)(17,45)(19,20)(29,41)(30,55)(39,56)(40,57)(42,61)(44,47)(46,48)(54,60)(59,64)(62,63)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72312375, #72312387, #72312573, … |
| maximal normal subgroup of | #72312375, #72312387, #72312573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536248);