Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139604 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
584 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(2,5)(3,6)(7,8)(9,10)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(1,2,6,8,4,7,3,5)(11,18)(14,23)(15,20)(19,21);(2,7)(5,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=cgi,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72313413, #72313423, #72313440, … |
| maximal normal subgroup of | #72313413, #72313423, #72313440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46813);