Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(9,10,13,17,14,18,12,21)(11,15,19,16,20,24,22,23);(1,2)(3,4)(5,6,7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,17)(18,24)(21,23);(1,3,4,2)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210005, #72235145, #72235150, … |
| maximal normal subgroup of | #72210005, #72235145, #72235150, … |
| maximal quotient of | #72210000, C2×#123670, #72235192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30866);