Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511799 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #23910 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1234 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,21)(14,18)(16,20)(19,24)(22,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(11,13)(12,20)(16,18)(17,23)(21,24);(9,11)(12,19)(13,14)(15,16)(17,21)(20,22)Polycyclic presentation:
2^9|c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72219496, #72219527, #72238413 |
| maximal normal subgroup of | #72219496, #72219527, #72238413 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419325);