Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | #34964 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(1,2,3,4)(14,19)(15,23)(16,25)(18,24)(20,27)(21,28);(6,9)(10,12)(13,15,18,23,17,16,24,25)(14,20,22,28,19,21,26,27);(13,16,18,25,17,15,24,23)(14,21,22,27,19,20,26,28);(5,7,8,11)(6,10,9,12)(15,16)(20,21)(23,25)(27,28)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257264, #72257277, #72263878, … |
| maximal normal subgroup of | #72257264, #72257277, #72263878, … |
| maximal quotient of | #72264864 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375113);