Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
5109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,27)(23,24,28,26);(1,2,3,4)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(7,11)(10,12)(15,19)(17,20)(22,27)(24,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,e^2=f,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 | #72232582, #72232628, #72232686, … |
| maximal normal subgroup of | #72232582, #72232628, #72232686, … |
| maximal quotient of | #72278515, #72278758, C2×#2171279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2078475);