Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2630 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)(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,21,23,18)(15,19,22,27,24,26,28,25);(2,4)(14,18)(15,22)(16,23)(20,21)(24,28)(25,27);(6,9)(10,12)(13,15,16,22,17,24,23,28)(14,19,20,27,21,26,18,25);(13,15,23,28,17,24,16,22)(14,19,18,25,21,26,20,27);(5,7,8,11)(6,10,9,12)(15,24)(19,26)(22,28)(25,27)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72224791, #72281706, #72281954, … |
| maximal normal subgroup of | #72224791, #72281706, #72281954, … |
| maximal quotient of | #72224783 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374754);