Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1497 conjugacy classes of subgroups; only the 246 normal subgroups are shown.
This group has 1497 conjugacy classes of subgroups — too many to draw. Only its 246 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:
(1,2,5,13,17,28,3,15)(4,20,23,30,21,9,18,26)(6,14,16,8,7,10,19,11)(12,31,29,22,27,32,25,24);(2,8,14,13)(3,16)(4,21)(5,19)(9,25,12,30)(10,15,28,11)(18,31)(20,29,27,26)(22,24)(23,32);(2,9)(3,5)(4,22)(8,25)(10,27)(11,29)(12,14)(13,30)(15,26)(16,19)(18,32)(20,28)(21,24)(23,31);(2,10)(8,15)(9,27)(11,13)(12,20)(14,28)(25,26)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213685, #72215564, #72216603, … |
| maximal normal subgroup of | #72213685, #72215564, #72216603, … |
| maximal quotient of | C2×#200876, #72240357, #72284774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108072);