Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11753 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | #2021 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,11)(4,8)(5,17)(6,10)(7,13,21,29)(9,19)(12,26)(14,24)(15,25)(16,23)(18,22)(20,27,31,30)(28,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,24)(10,21)(11,17)(15,27)(16,28)(18,30)(19,29)(22,32)(23,31)(25,26);(1,4,6,16)(2,8,10,23)(3,12,14,28)(5,18,19,15)(7,20,21,31)(9,25,17,22)(11,26,24,32)(13,27,29,30)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,h^b=hi,d^c=dhi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72283813, C2×#150394, #72284593, … |
| maximal normal subgroup of | #72283813, C2×#150394, #72284593, … |
| maximal quotient of | C2×#150394, #72284996, #72285439 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57590);