Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508074 |
| 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 | #52307 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 901 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,7)(10,14)(11,17)(12,20)(15,23)(16,18)(21,24);(1,2)(3,5)(4,6)(7,8)(10,15)(12,20)(14,23)(16,21)(18,24)(19,22);(1,3,5,2)(4,6,8,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^9|c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72271403 |
| maximal normal subgroup of | #72271403 |
| maximal quotient of | #72271427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415434);