Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
644 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,14,10,20,18,13,11,16,8,19,9,15,17,12);(1,2,3,4)(6,12)(7,17)(8,18)(9,10)(13,16)(14,15)(19,20);(6,13)(7,18)(8,17)(9,10)(12,16)(14,20)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,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 | #72250879, #72250947, #72250956, … |
| maximal normal subgroup of | #72250879, #72250947, #72250956, … |
| maximal quotient of | #72265082, #72265096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44588);