Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10018 |
| commutator, Frattini, cosocle | C4×M16 | C23 |
| prime radical | #575 | D4 |
| LCS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,8,4)(3,13,18,31)(5,20,22,29)(6,12,7,11)(9,25,14,15)(10,24,19,23)(16,28,17,30)(21,26,32,27);(1,3)(2,9)(4,14)(5,21)(6,16)(7,17)(8,18)(10,28)(11,26)(12,27)(13,20)(15,23)(19,30)(22,32)(24,25)(29,31);(1,4,8,2)(3,14,18,9)(5,20,22,29)(6,11,7,12)(10,24,19,23)(13,21,31,32)(15,30,25,28)(16,26,17,27)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=egh,g^c=gi,h^c=hi,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72231251, C2×#143165, #72306763, … |
| maximal normal subgroup of | #72231251, C2×#143165, #72306763, … |
| maximal quotient of | C2×#143165, #72309249 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50361);