Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152670 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #23875 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3499 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
581 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 581 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,11)(3,13,16,27)(4,19,7,12)(5,10,21,25)(8,22,23,20)(9,26,28,17)(14,24,18,29)(15,30,32,31);(2,8,23,11,12,25,10,19)(4,20)(5,22)(6,21)(9,24,13,31,30,27,29,28)(14,32)(15,17)(16,26);(1,3)(2,9)(4,16)(5,17)(6,14)(7,18)(8,24)(10,29)(11,31)(12,30)(13,23)(15,22)(19,28)(20,26)(21,32)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72283303, #72283528 |
| maximal normal subgroup of | #72283303, #72283528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59866);