Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485328 |
| 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 | C2×C4⋊C4 | C24 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,26)(20,23)(21,29)(22,24)(25,28)(27,32)(30,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,23)(19,21)(20,28)(24,25)(26,31)(29,32);(17,19,22,21)(20,27,28,30)(23,25)(31,32);(2,7)(6,15)(8,12)(9,16)(18,24)(23,25)(26,29)(31,32)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72240300, #72261335, #72278295, … |
| maximal normal subgroup of | #72240300, #72278295, #72278872 |
| maximal quotient of | #72240576 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392560);