Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #25763 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,21)(17,20)(18,22)(23,27)(24,26)(25,28);(6,10)(11,12)(13,15,17,25)(14,19,20,28)(16,23,24,18)(21,27,26,22);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,20)(15,16)(18,23)(19,26)(21,28)(24,25);(5,8,9,7)(6,12,10,11)(13,16)(14,21)(15,23)(17,24)(18,25)(19,27)(20,26)(22,28);(15,24)(16,25)(18,23)(19,26)(21,28)(22,27)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256862, #72256912, #72256954, … |
| maximal normal subgroup of | #72256862, #72256912, #72256954, … |
| maximal quotient of | #72265209, #72265257 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492038);