Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,16,11,14,10,15)(7,17,21,32,19,12,20,27)(8,13,23,31,24,29,25,30)(18,26,35,28,34,33,22,36);(2,4)(5,7,10,21,11,19,9,20)(6,12,15,27,14,17,16,32)(8,22,23,34,24,35,25,18)(13,28,31,26,29,36,30,33);(5,8)(6,13)(7,18,19,34)(9,23)(10,25)(11,24)(12,26,17,33)(14,29)(15,30)(16,31)(20,22,21,35)(27,28,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210978, #72210987, #72239632, … |
| maximal normal subgroup of | #72210978, #72210987, #72239632, … |
| maximal quotient of | #72249751, C2×#123913, #72316684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31109);