Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11753 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | #2021 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 492 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:
(1,2,5,9,6,10,18,26)(3,11,13,27,14,23,30,32)(4,15,17,31,16,8,22,24)(7,19,20,12,21,28,25,29);(1,3)(2,7)(4,12)(5,13)(6,14)(8,23)(9,25)(10,21)(11,15)(16,29)(17,19)(18,30)(20,26)(22,28)(24,32)(27,31);(4,16)(8,15)(17,22)(24,31)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=i,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×#150395, #72285232, … |
| maximal normal subgroup of | #72283813, C2×#150395, #72285232, … |
| maximal quotient of | C2×#150395, #72285169, #72285374 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57591);