Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
4383 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)(7,12)(8,11)(9,10)(13,14)(15,18)(16,20)(17,19)(21,22)(23,25)(24,27)(26,28);(6,10)(7,11)(13,15,17,20)(14,16,19,18)(22,23)(24,28);(7,11)(8,12)(15,20)(16,18)(21,22,25,23)(24,27,28,26);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,22)(24,27,28,26);(5,8)(6,11)(7,10)(9,12)(13,16,17,18)(14,15,19,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232521, #72232535, #72232540, … |
| maximal normal subgroup of | #72232521, #72232535, #72232540, … |
| maximal quotient of | #72329873, #72330478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990451);