Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,13,25,29)(7,15,17,31,18,12,23,28)(8,19,22,30,21,11,24,27)(16,26,33,35,20,32,34,36);(1,2,3,4)(6,11)(12,26)(13,19)(14,30)(15,32)(27,29)(28,36)(31,35);(5,7,10,18)(6,12,13,15)(8,20,21,16)(9,23,25,17)(11,26,19,32)(14,31,29,28)(22,33,24,34)(27,36,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,e^b=eh,f^b=fh,d^c=dhi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53813);