Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144629 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,7,9,23)(3,11)(4,16)(8,24,25,22)(10,12,27,29)(14,15)(17,31,32,26)(19,20);(2,8,9,25)(4,16)(7,22,23,24)(10,17,27,32)(12,26,29,31)(13,28)(18,30)(19,20);(1,2)(3,12)(4,17)(5,9)(6,10)(7,15)(8,20)(11,23)(13,22)(14,29)(16,25)(18,26)(19,32)(21,27)(24,30)(28,31)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fghi,c^b=cf,e^b=eh,f^b=fgi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268098 |
| maximal subgroup of | #72209704, #72209705, #72209706, … |
| maximal normal subgroup of | #72209704, C2×#144629, #72235018, … |
| maximal quotient of | #72210066, C2×#144629, #72266617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51825);