Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,11)(15,19)(18,20)(21,22)(23,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(21,22,25,28)(23,24,27,26);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,20,19,18)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297915, #72297916, #72297930, … |
| maximal normal subgroup of | #72297915, #72297916, #72297930, … |
| maximal quotient of | #72298181, #72298543, #72330825 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1274687);