Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17)(8,21,9,14)(10,15)(11,16)(12,18,28,22)(13,23,30,24)(19,26)(20,33)(25,32)(27,34,31,35)(29,36);(1,2,3,4)(5,7)(6,12,16,28)(8,22)(9,18)(10,19)(11,20)(13,29,30,26)(14,17,21,33)(15,31,32,27)(23,35)(24,34)(25,36);(5,8)(6,13)(7,18)(9,11)(10,24)(12,26)(14,15)(16,30)(17,27)(19,35)(20,22)(21,32)(23,25)(28,29)(31,33)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,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, 53822);