Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3398 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,23)(24,26,28,27);(1,2,3,4)(6,7)(10,11)(13,14,17,16)(15,18,20,19)(22,23)(26,27);(1,2)(3,4)(8,12)(10,11)(14,16)(18,19)(21,22,25,23)(24,27,28,26);(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,18,16,19)(21,23,25,22)(24,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,19,20,18)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=fi,d^2=f,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 | #72300646, #72300661, #72300664, … |
| maximal normal subgroup of | #72300646, #72300661, #72300664, … |
| maximal quotient of | #72323652, #72328887, #72328890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2188120);