Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94674 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7412 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1961 | C22 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1961 ◁ G | |
| LECS | G ▷ #1961 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:
(1,2)(3,12,14,25)(4,17,19,27)(5,10,21,9)(6,11)(7,13,22,29)(8,20,26,32)(15,28,16,31)(18,24)(23,30);(1,3,5,14,6,15,21,16)(2,7,10,24,11,22,9,23)(4,18,19,13,20,30,32,29)(8,25,17,28,27,31,26,12)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fhi,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,h^a=hi,c^b=cf,d^b=defi,e^b=eh,f^b=fi,d^c=dgh,e^c=ei,e^d=ehi,f^d=fi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1870);