Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1578 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 278 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 — 4 generators on 24 points:
(9,10)(11,15)(12,13)(14,16)(17,18)(19,23,24,21)(20,22);(1,2,4,6)(3,7,8,5)(10,13)(14,15)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,5,6,7)(17,19)(18,21)(20,24)(22,23);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^9|a^2=eh,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282131, #72282185, #72282239 |
| maximal normal subgroup of | #72282131, #72282185, #72282239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322788);