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 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 694 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)(3,13)(4,9)(5,18)(6,10)(7,12)(8,15)(11,20)(14,29)(16,30)(17,28)(19,23)(21,26)(22,27)(24,31)(25,32);(1,3)(2,8)(4,14)(5,19)(6,22)(7,16)(9,23)(10,24)(11,27)(12,25)(13,18)(15,20)(17,32)(21,31)(26,30)(28,29);(1,4,7,6)(2,9,12,10)(3,14,16,22)(5,20,21,17)(8,23,25,24)(11,18,28,26)(13,29,30,27)(15,31,32,19);(2,10,12,9)(5,21)(8,24,25,23)(11,18,28,26)(13,29,30,27)(15,32)(17,20)(19,31)Polycyclic presentation:
2^9|c^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 | #72231251, #72245189, #72250033, … |
| maximal normal subgroup of | #72231251, #72245189, #72250033, … |
| maximal quotient of | #72250372, #72250947, C2×#485587 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392783);