Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23906 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,22)(19,23)(21,24);(5,8)(10,14)(12,20)(15,22)(16,17)(18,24)(21,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22)Polycyclic presentation:
2^9|c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,d^c=dg,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214408, #72219501, #72219740, … |
| maximal normal subgroup of | #72214408, #72219501, #72219740, … |
| maximal quotient of | #72250224, C2×#144387, #72316296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51583);