Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2316 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)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(6,9)(7,11)(13,16,15,22,18,25,17,24)(14,20,26,27,21,28,19,23);(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,23,28,27);(6,9)(7,11)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257264, #72257391, #72263875, … |
| maximal normal subgroup of | #72257264, #72257391, #72263875, … |
| maximal quotient of | #72264904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513373);