Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149901 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #23909 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
627 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 627 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,35)(8,13)(9,25)(10,15,27,33)(11,16)(12,19,30,18)(14,26,28,24)(21,34,22,29)(23,32,36,31);(5,7,8,23,11,22,9,18)(6,12,28,35)(10,20,24,36,27,21,26,19)(13,17,15,31)(14,34,16,32)(25,29,33,30);(2,4)(5,8,11,9)(6,13,16,25)(7,18,22,23)(10,24,27,26)(12,29,32,17)(14,33,28,15)(19,21,36,20)(30,34,31,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72306380, #72306382 |
| maximal normal subgroup of | #72306380, #72306382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59849);