Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,18,22,28)(8,14)(9,15)(10,16)(11,17)(12,20,13,21)(19,30,35,36)(23,29,34,27)(24,31)(25,32)(26,33);(5,7)(6,12)(8,23)(9,21)(10,20)(11,22)(13,17)(14,30)(15,28)(16,18)(19,24)(25,35)(26,34)(27,31)(29,32)(33,36);(1,2,4,3)(5,7)(6,13,17,12)(8,19)(9,20)(10,21)(11,22)(14,29,33,27)(15,28,16,18)(23,24)(25,34)(26,35)(30,31,36,32);(1,3,4,2)(5,8,10,25,11,26,9,24)(6,14,16,32,17,33,15,31)(7,19,21,34,22,35,20,23)(12,27,28,36,13,29,18,30)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293923);