Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629016 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,27)(17,22)(18,23)(19,24)(20,26)(21,28);(5,7,8,11)(6,9,10,12)(13,15,17,26,19,27,18,21)(14,20,23,25,24,28,22,16);(6,10)(9,12)(13,14)(15,20)(16,21)(17,23)(18,22)(19,24)(25,26)(27,28);(2,4)(13,16,17,20,19,25,18,28)(14,21,23,15,24,26,22,27);(13,17,19,18)(14,22,24,23)(15,21,27,26)(16,20,25,28)Polycyclic presentation:
2^9|b^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250074, #72250109, #72250121, … |
| maximal normal subgroup of | #72250074, #72250109, #72250121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536954);