Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
4183 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)(6,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,23)(24,26,28,27);(5,6,8,9)(7,10,11,12)(13,14,17,16)(15,18,20,19)(22,23)(26,27);(7,11)(10,12)(14,16)(18,19)(21,22,25,23)(24,27,28,26);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,25,22)(24,26,28,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,18,20,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72230894, #72232382, #72232402, … |
| maximal normal subgroup of | #72230894, #72232382, #72232402, … |
| maximal quotient of | #72278194, #72278526, #72278799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3272667);