Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C23 | #428 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 235 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,20,26,24,30,32,25,22,27,21,31,23,19,28,29);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19,22,30)(18,24,27,23)(20,26,21,31)(25,32,29,28);(2,7)(6,12)(8,14)(13,16)(17,20,23,28,22,21,24,32)(18,25,30,31,27,29,19,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=cgh,d^a=di,d^b=dg,e^b=egi,g^b=gi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287105, #72287179, #72287209, … |
| maximal normal subgroup of | #72287105, #72287179, #72287209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36329);