Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 32 points:
(1,2,6,12,4,14,3,15)(5,11,17,29,19,9,21,27)(7,13,22,8,16,10,18,25)(20,26,31,23,30,28,32,24);(1,3,16,22)(2,8,10,15)(4,6,7,18)(5,17,30,32)(9,23,26,27)(11,24,28,29)(12,14,25,13)(19,21,20,31);(2,9,10,26)(5,20)(8,23,15,27)(11,13,28,14)(12,29,25,24)(17,31)(19,30)(21,32)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=dh,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eg,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72260536, #72260654 |
| maximal normal subgroup of | #72260536, #72260654 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42747);