Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| 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 | #1867 |
| 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 | |
997 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)(8,24,9,15)(10,16)(11,27)(12,17)(13,20,30,19)(14,25,32,26)(21,33)(22,28)(23,34)(29,36,31,35);(5,7,11,22,12,23,10,21)(6,13,16,31,17,30,27,29)(8,19,25,35,9,20,26,36)(14,28,15,18,32,33,24,34);(2,4)(5,8)(6,14)(7,19)(9,12)(10,26)(11,25)(13,28)(15,16)(17,32)(18,31)(20,23)(21,36)(22,35)(24,27)(29,34)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313916, #72314013, #72314023 |
| maximal normal subgroup of | #72313916, #72314013, #72314023 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53823);