Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1812 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8609 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,14)(4,20)(5,11,24,9)(6,26)(7,13)(8,17)(10,15)(12,21)(16,39)(18,36,44,57)(19,48)(22,50,53,49)(23,51)(25,32,52,35)(27,54,30,46)(28,56)(29,43)(31,41,33,59)(34,42)(37,40)(38,45)(47,63,62,60)(55,61,58,64);(1,3,15,40,6,19,23,43,7,16,42,56,21,45,4,17)(2,8,20,38,12,28,34,39,13,29,51,48,26,37,10,14)(5,18,41,61,25,47,53,54,24,44,59,64,52,62,22,46)(9,30,49,63,32,58,33,36,11,27,50,60,35,55,31,57);(2,9)(3,16)(5,24)(8,27)(10,31)(11,13)(12,32)(14,36)(17,43)(19,45)(20,49)(22,53)(25,52)(26,35)(28,55)(29,30)(33,34)(37,58)(38,60)(39,57)(40,56)(41,59)(48,63)(50,51)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=ei,f^c=fhi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287254, #72287291, #72321074, … |
| maximal normal subgroup of | #72287254, #72287291, #72321074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41972);