Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2129 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)(3,4)(5,6,11,16)(7,17,21,31)(8,14,9,13)(10,25,26,15)(12,23,29,19)(18,33,20,34)(22,30,24,32)(27,36,28,35);(5,7)(6,12)(8,19)(9,23)(10,18)(11,21)(13,17)(14,31)(15,27)(16,29)(20,26)(22,35)(24,36)(25,28)(30,33)(32,34);(2,4)(7,18)(10,26)(12,27)(15,25)(17,33)(19,35)(20,21)(22,24)(23,36)(28,29)(30,32)(31,34);(5,8,11,9)(6,13,16,14)(7,19,21,23)(10,22,26,24)(12,17,29,31)(15,30,25,32)(18,35,20,36)(27,33,28,34)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406103);