Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| 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×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18,22,34)(8,14,24,33)(9,15)(10,16)(11,17)(12,20)(13,23,32,26)(19,30)(21,27,25,31)(28,36)(29,35);(5,7,10,21,11,22,9,25)(6,12,16,29,17,30,15,28)(8,20,23,35,24,19,26,36)(13,31,14,18,32,27,33,34);(2,4)(5,8)(6,13)(7,19)(9,26)(10,23)(11,24)(12,27)(14,16)(15,33)(17,32)(18,28)(20,22)(21,36)(25,35)(29,34)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=ci,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72337451, #72337454, #72337464, … |
| maximal normal subgroup of | #72337451, #72337454, #72337464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53834);