Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1752 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,28,43,8,11,29,39,5,12,26,41)(3,15,18,44,17,32,47,42,19,40,27,55,25,9,30,33)(4,10,22,37,24,38,54,63,23,35,53,60,21,36,51,61)(16,34,45,59,20,48,49,64,46,58,56,62,50,31,52,57);(1,3)(2,9)(4,20,23,50)(5,25)(6,27)(7,17)(8,19)(10,34,35,58)(11,32)(12,40)(13,42)(14,15)(16,24,46,21)(18,29)(22,52,53,49)(26,47)(28,30)(31,38,48,36)(33,39)(37,62,60,59)(41,44)(43,55)(45,51,56,54)(57,61,64,63);(1,4,6,22,7,24,28,54,8,23,29,53,5,21,26,51)(2,10,13,37,14,38,43,63,11,35,39,60,12,36,41,61)(3,16,18,45,17,20,47,49,19,46,27,56,25,50,30,52)(9,31,33,57,15,34,44,59,32,48,42,64,40,58,55,62);(2,11)(9,32)(10,35)(12,14)(13,39)(15,40)(31,48)(33,42)(34,58)(36,38)(37,60)(41,43)(44,55)(57,64)(59,62)(61,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,d^a=di,c^b=ch,e^b=ei,f^b=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319756, #72319772, #72319774, … |
| maximal normal subgroup of | #72319756, #72319772, #72319774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400178);