Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(14,18)(15,19)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(22,23)(26,27);(7,11)(10,12)(14,18)(16,20)(21,22,25,23)(24,27,28,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23,25,22)(24,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232348, #72232362, #72232412, … |
| maximal normal subgroup of | #72232348, #72232362, #72232412, … |
| maximal quotient of | #72251938, #72278753, #72278805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513709);