Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
4211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,25,26)(23,24,28,27);(2,4)(6,9)(10,11)(13,15,17,20)(14,16,18,19)(21,22)(23,27)(24,28)(25,26);(1,2)(3,4)(7,12)(10,11)(14,18)(15,20)(22,26)(23,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232300, #72232348, #72232383, … |
| maximal normal subgroup of | #72232300, #72232348, #72232383, … |
| maximal quotient of | #72298433, #72323716, #72328605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1787641);