Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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×#3297 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,25)(14,19,22,28)(16,24,17,23)(20,27,21,26);(14,20)(15,23)(16,17)(21,22)(24,25)(26,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(17,18)(19,26)(20,21)(23,24)(27,28);(6,11)(10,12)Polycyclic presentation:
2^9|b^2=fi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72257369, #72263880, … |
| maximal normal subgroup of | #72239609, #72257369, #72263880, … |
| maximal quotient of | #72239705, #72264528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425112);