Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3391 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
776 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,7,4,3,5,8)(9,10,12,15,14,18,11,19,13,17,20,23,21,24,22,16);(5,6)(7,8)(10,15)(11,20)(12,22)(14,21)(16,18)(17,23)(19,24);(1,3,4,2)(5,7,6,8)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=egi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72342415, #72342417, #72342424, … |
| maximal normal subgroup of | #72342415, #72342417, #72342424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39100);