Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486333 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18194 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 899 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 — 4 generators on 32 points:
(1,2)(3,12)(4,7)(5,19)(6,11)(8,13)(9,16)(10,18)(14,23)(15,26)(17,21)(20,27)(22,30)(24,29)(25,28)(31,32);(2,7)(3,13)(4,16)(5,18)(8,23)(9,10)(11,21)(12,20)(14,29)(15,28)(17,30)(19,22)(24,25)(26,31);(4,17)(14,29)(16,30)(27,32);(1,3,5,13,6,15,18,28)(2,8,10,12,11,25,19,26)(4,14,16,27,17,29,30,32)(7,20,22,23,21,31,9,24)Polycyclic presentation:
2^9|d^2=g,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231603, #72231708, #72231710, … |
| maximal normal subgroup of | #72231603, #72231708, #72231710, … |
| maximal quotient of | #72250369, C2×#486333, #72279960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393529);