Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(14,15)(16,20)(21,22,26,24)(23,25,28,27);(2,4)(6,9)(7,12)(16,20)(18,19)(23,28)(25,27);(1,2,3,4)(6,9)(10,11)(13,14,17,15)(16,18,20,19)(21,23,26,28)(22,27,24,25);(5,7,8,12)(6,10,9,11)(13,15,17,14)(16,19,20,18)(21,24,26,22)(23,25,28,27);(5,6)(7,11)(8,9)(10,12)(13,16)(14,18)(15,19)(17,20)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fgi,c^2=fhi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300716, #72300725, #72300800, … |
| maximal normal subgroup of | #72300716, #72300725, #72300800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3992010);