Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,22)(10,13)(11,16)(12,18)(15,26)(19,33)(20,28)(21,31)(23,30)(24,32)(25,34)(27,35)(29,36);(1,3,4,2)(5,7)(6,12)(8,18)(9,23)(10,25)(11,20)(13,27)(14,28)(15,29)(16,30)(17,22)(19,26)(21,36)(24,35)(31,34)(32,33);(6,13)(10,26)(12,27)(14,31)(15,16)(17,33)(19,25)(21,24)(22,32)(28,34)(29,30)(35,36);(5,8,11,9)(6,14,16,22)(7,18,20,23)(10,21,26,24)(12,28,30,17)(13,31,15,32)(19,35,25,36)(27,34,29,33)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401579);