Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 136 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:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(2,5)(3,8)(6,7)(10,16)(14,20)(17,21)(22,24);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72342546, #72342594, #72342596 |
| maximal normal subgroup of | #72342546, #72342594, #72342596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398293);