Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512876 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #124 | C24 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1058 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:
(2,8)(4,5)(6,20)(9,24)(10,23)(11,17)(12,18)(13,30)(22,25)(26,32)(27,31)(28,29);(1,2)(3,9)(4,8)(5,17)(6,10)(7,11)(12,22)(13,31)(14,26)(15,27)(16,23)(18,32)(19,28)(20,29)(21,25)(24,30);(1,3)(2,9)(4,12)(5,18)(6,21)(7,15)(8,22)(10,25)(11,27)(13,19)(14,20)(16,30)(17,32)(23,24)(26,29)(28,31);(1,4,7,5)(2,8,11,17)(3,12,15,18)(6,16,20,19)(9,22,27,32)(10,23,29,28)(13,21,30,14)(24,26,31,25)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279638 |
| maximal normal subgroup of | #72279638 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420074);