Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2378 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)(7,11)(8,9)(10,12)(13,14,19,16,18,22,17,21)(15,23,27,28,26,20,24,25);(2,4)(6,9)(10,11)(14,16)(15,24)(17,19)(21,22)(25,28)(26,27);(5,7,8,12)(6,10,9,11)(13,15,19,27,18,26,17,24)(14,20,16,25,22,23,21,28);(13,16,19,22,18,21,17,14)(15,25,27,23,26,28,24,20);(15,26)(20,23)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238782, #72239652, #72239904, … |
| maximal normal subgroup of | #72238782, #72239652, #72239904, … |
| maximal quotient of | #72237292 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373394);