Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12759 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,7,13)(3,15,18,26)(4,19,21,16)(5,14,23,32)(6,12,24,31)(8,11,22,30)(9,29,20,27)(10,28,17,25);(1,3)(2,9)(4,11)(5,16)(6,19)(7,17)(8,27)(10,24)(12,20)(13,21)(14,28)(15,31)(18,22)(23,29)(25,30)(26,32);(1,4)(2,10)(3,11)(5,15)(6,25)(7,26)(8,28)(9,24)(12,18)(13,29)(14,27)(16,31)(17,32)(19,30)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=egh,f^b=fgh,g^b=gi,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59384);