Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,13,10,14,8,12)(7,15,16,11,18,19,17,20)(21,22,24,30,34,36,33,27,26,32,23,29,25,31,35,28);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,15)(14,20)(22,27)(23,33)(24,35)(25,34)(28,32)(29,31)(30,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(22,28)(23,33)(24,35)(25,34)(27,32)(29,36)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=ghi,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44161);