Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
3711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,8)(7,11,12,10)(14,18)(15,19)(21,22)(23,28)(24,26)(25,27);(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,16,19,20)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(21,22)(23,28)(24,26)(25,27);(5,8,9,6)(7,11,12,10)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,19)(15,18)(17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,d^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 | #72232630, #72232642, #72232707, … |
| maximal normal subgroup of | #72232630, #72232642, #72232707, … |
| maximal quotient of | #72298544, #72330044, #72330532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1220886);