Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485043 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #23984 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1356 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1356 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,8)(6,10)(7,13)(9,18)(12,25)(14,23)(16,21)(17,28)(19,22)(20,26)(24,30)(27,32)(29,31);(1,3,5,14)(2,7,8,21)(4,16,18,13)(6,12,19,27)(9,23,15,11)(10,20,22,31)(17,29,30,26)(24,32,28,25);(3,12)(7,20)(9,24)(13,26)(14,27)(15,28)(16,29)(21,31);(2,8)(7,21)(9,15)(10,22)(11,23)(20,31)(24,28)(25,32)Polycyclic presentation:
2^9|b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,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 | #72227564, #72227614, #72231849, … |
| maximal normal subgroup of | #72227564, #72231851, #72234736, … |
| maximal quotient of | #72227595, #72237736, #72237846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392277);