Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C24 |
| socle | C2 | C2×#2196 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1045 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1045 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:
(2,9)(3,14)(5,18)(6,17)(11,21)(12,29)(13,19)(15,32)(16,22)(20,23)(24,27)(26,31);(1,2)(3,10)(4,11)(5,19)(6,21)(7,12)(8,13)(9,18)(14,31)(15,25)(16,28)(17,29)(20,27)(22,26)(23,30)(24,32);(1,3)(2,10)(4,15)(5,20)(6,22)(7,23)(8,16)(9,24)(11,25)(12,30)(13,28)(14,17)(18,32)(19,27)(21,26)(29,31);(1,4,8,7)(2,11,13,12)(3,15,16,23)(5,17,18,6)(9,21,19,29)(10,25,28,30)(14,32,22,20)(24,26,27,31)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gh,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,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 | #72283303, #72283371, #72283393, … |
| maximal normal subgroup of | #72283303, #72283371, #72283393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420089);