Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,20)(11,21,13,18)(12,17,22,16)(14,24,23,19);(5,6)(7,8)(9,11,15,13)(10,16,20,17)(12,23,22,14)(18,24,21,19);(2,4)(6,8)(9,12)(10,17)(11,14)(13,23)(15,22)(16,20)(18,19)(21,24);(5,6)(7,8)(9,13,15,11)(10,18,20,21)(12,23,22,14)(16,24,17,19);(6,8)(11,13)(12,22)(16,17)(18,21)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72281486, #72281600, #72282243, … |
| maximal normal subgroup of | #72281486, #72281600, #72282243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480804);