Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4174 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)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(19,20)(22,23);(1,2,4,3)(9,11)(10,12)(13,16)(14,15)(17,18,21,24)(19,22,20,23);(2,3)(10,14)(12,15)(17,19,21,20)(18,22,24,23);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,3,4,2)(9,12)(10,11)(13,15)(14,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,c^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72330809, #72330826, #72330914, … |
| maximal normal subgroup of | #72330809, #72330826, #72330914, … |
| maximal quotient of | #72333005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2034624);