Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128009 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
860 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 860 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(2,7,9,19)(3,12)(4,15)(8,22,23,32)(10,21,11,20)(13,28)(14,27)(16,30)(18,24,31,25)(26,29);(1,2)(3,8)(4,7,16,20)(5,10)(6,11)(9,17)(12,25)(13,18,26,22)(14,24)(15,21,30,19)(23,27)(28,31,29,32);(1,3,5,14)(2,8,10,24)(4,13,15,29)(6,12,17,27)(7,18,21,32)(9,23,11,25)(16,26,30,28)(19,31,20,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233041, #72233046, #72233048, … |
| maximal normal subgroup of | #72233041, #72233046, #72233048, … |
| maximal quotient of | C2×#128050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35246);