Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152194 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12759 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #61110 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 822 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 32 points:
(1,2,5,12)(3,14,17,25)(4,11,6,8)(7,13,22,26)(9,15,28,19)(10,20,30,23)(16,27,24,32)(18,31,21,29);(2,8)(4,5)(6,23)(9,25)(10,13)(11,30)(14,31)(15,24)(16,17)(19,21)(20,22)(27,29);(1,3)(2,9)(4,19)(5,21)(6,24)(7,18)(8,25)(10,29)(11,31)(12,32)(13,27)(14,30)(15,23)(16,20)(17,22)(26,28)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=eghi,f^b=fgh,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72245262, #72245391, #72245414, … |
| maximal normal subgroup of | #72245262, #72245391, #72245414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59390);