Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1850 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1850 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(14,20)(21,24);(1,3)(2,5)(4,8)(6,7)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^b=ei,f^b=fh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214228, #72214493, #72214512, … |
| maximal normal subgroup of | #72214228, #72214493, #72214512, … |
| maximal quotient of | C2×#108225, #72280984, #72311433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15421);