Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,17,21,28,19,29,20,12)(8,22,24,32,26,13,25,31)(18,33,35,36,34,30,23,27);(1,2)(3,4)(5,7)(6,12,16,28)(8,23,26,35)(9,20)(10,21)(11,19)(13,30)(14,29,15,17)(18,25,34,24)(22,33)(27,31)(32,36);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,32)(16,22)(17,30)(19,34)(20,35)(21,23)(28,36)(29,33);(6,14,16,15)(7,19)(12,17,28,29)(13,31,22,32)(18,34)(20,21)(23,35)(27,30,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280664, #72318910, #72318940, … |
| maximal normal subgroup of | #72280664, #72318910, #72318940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273192);