Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485043 |
| 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×C22⋊C4 | C24 |
| prime radical | #18162 | 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 | |
1044 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,8)(4,7)(5,9)(6,11)(10,16)(12,25)(13,22)(14,19)(15,23)(17,21)(18,20)(24,30)(26,28)(27,32)(29,31);(2,7)(3,12)(4,16)(8,21)(9,10)(11,20)(14,29)(15,27)(17,30)(18,28)(19,25)(22,26)(23,31)(24,32);(1,3,13,27)(2,8,22,32)(4,17,28,30)(5,15,6,12)(7,19,26,31)(9,23,11,25)(10,24,20,21)(14,16,29,18);(2,9)(7,10)(8,23)(11,22)(19,24)(20,26)(21,31)(25,32)Polycyclic presentation:
2^9|c^2=gi,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 | #72240214, #72338692, #72338800, … |
| maximal normal subgroup of | #72240214, #72338692, #72338800, … |
| maximal quotient of | #72240609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392318);