Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12117 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1595 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #9041 |
| prime radical | #33530 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,14,5,11,8,16,26,12,18,15,19,32)(3,17,22,9,23,10,4,25,24,31,21,30,27,28,20,29);(1,3,18,23,8,24,7,27)(2,9,15,25,16,30,14,29)(4,19,21,26,20,5,22,6)(10,13,31,32,28,12,17,11);(1,4)(2,10)(3,19)(5,24)(6,27)(7,22)(8,20)(9,13)(11,29)(12,30)(14,17)(15,31)(16,28)(18,21)(23,26)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217920, #72219620, #72224559, … |
| maximal normal subgroup of | #72217920, #72219620, #72224559, … |
| maximal quotient of | #72216739, #72216744, #72216747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44590);