Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| 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 | |
1533 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)(7,15,19,11)(8,20,22,12)(9,14)(10,13)(16,26)(17,27)(18,31,23,28)(21,32,24,29)(25,30)(33,35)(34,36);(1,2,4,3)(5,7)(6,11)(8,17)(9,23)(10,19)(12,27)(13,15)(14,31)(16,21)(18,25)(20,34)(22,36)(24,35)(26,29)(28,30)(32,33);(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,20)(14,32)(15,33)(17,23)(18,36)(19,35)(21,25)(27,31)(28,34)(29,30);(1,3,4,2)(5,8)(6,12)(7,17)(9,21)(10,22)(11,27)(13,20)(14,29)(15,34)(16,23)(18,35)(19,36)(24,25)(26,31)(28,33)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268676);