Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138944 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9259 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#576 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,36,32,12,7,15,35,10,25,14,5,30)(3,16,20,54,61,45,33,53,22,55,31,51,21,62,19,59)(4,11,27,44,29,64,26,38,28,48,34,50,24,47,17,46)(9,37,41,18,43,57,39,63,42,56,49,58,52,60,40,23);(1,3,17,56)(2,9,38,59)(4,23,35,19)(5,31,29,58)(6,33,28,63)(7,22,26,37)(8,21,27,57)(10,39,48,62)(11,45,13,40)(12,49,64,16)(14,43,44,54)(15,42,46,53)(18,32,20,24)(25,61,34,60)(30,41,47,55)(36,52,50,51);(2,10)(4,24)(5,6)(8,25)(9,39)(11,46)(12,36)(13,15)(14,30)(16,51)(18,23)(19,20)(21,61)(27,34)(28,29)(31,33)(32,35)(38,48)(40,42)(41,43)(44,47)(45,53)(49,52)(50,64)(54,55)(57,60)(58,63)(59,62)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72281073 |
| maximal normal subgroup of | #72281073 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46146);