Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| socle | C2 | C2×#2196 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(5,17)(7,18)(15,28)(19,29)(20,21)(30,31);(1,2)(3,7)(4,8)(5,9)(6,10)(11,19)(12,18)(13,20)(14,21)(15,22)(16,23)(17,24)(25,29)(26,30)(27,31)(28,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,19)(9,18)(10,21)(13,17)(15,25)(16,27)(20,24)(22,29)(23,31)(26,28)(30,32);(1,4,6,16)(2,8,10,23)(3,11,14,27)(5,15,17,28)(7,19,21,31)(9,22,24,32)(12,25,13,26)(18,29,20,30)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,g^a=gh,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279437, #72283143, #72283193, … |
| maximal normal subgroup of | #72279437, #72283143, #72283193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420053);