Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130854 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8376 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61259 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
578 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,12,8,13,3,15,7,14,18,30,22,31,17,27)(4,9,20,25,21,26,23,32,19,10,24,28,16,29,5,11);(1,3)(2,9)(4,5)(6,8)(7,17)(10,14)(11,30)(12,32)(13,29)(15,25)(16,20)(18,22)(19,23)(21,24)(26,31)(27,28);(1,4,7,19)(2,10)(3,5,17,23)(6,24,18,20)(8,21,22,16)(9,14)(11,15)(12,28)(13,29)(25,30)(26,31)(27,32)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231599, #72231701, … |
| maximal normal subgroup of | #72231596, #72231599, #72231701, … |
| maximal quotient of | C2×#130854, #72250852, #72279588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38050);