Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18152 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| prime radical | #2010 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 32 points:
(1,2)(3,12)(4,16)(5,8)(6,11)(7,13)(9,20)(10,19)(14,28)(15,29)(17,27)(18,24)(21,25)(22,32)(23,31)(26,30);(1,3,6,15)(2,7,11,23)(4,17,18,32)(5,21,19,14)(8,22,10,27)(9,25,26,28)(12,24,29,16)(13,30,31,20);(4,18)(13,31)(17,32)(20,30);(2,8,11,10)(4,18)(7,22,23,27)(9,16,26,24)(12,28,29,25)(13,31)(17,32)(20,30)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72281886, #72306550, #72306763, … |
| maximal normal subgroup of | #72281886, #72306550, #72306763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392782);