Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144631 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10390 |
| commutator, Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,5,9)(3,11,13,7)(4,15,16,23)(6,10,18,24)(8,21,22,17)(12,25,27,30)(14,26,29,20)(19,28,31,32);(1,3,5,13)(2,7,9,11)(4,12,16,27)(6,14,18,29)(8,19,22,31)(10,20,24,26)(15,30,23,25)(17,28,21,32);(1,4,6,17)(2,8,10,15)(3,12,14,28)(5,16,18,21)(7,19,20,30)(9,23,24,22)(11,25,26,31)(13,27,29,32)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72268800, #72340882, #72340884, … |
| maximal normal subgroup of | #72268800, #72340882, #72340884, … |
| maximal quotient of | #72268863 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51828);