Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | #25785 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2125 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)(5,6,9,10)(7,11,8,12)(15,23)(16,19)(20,25)(24,26);(6,10)(11,12)(13,14,17,22)(15,24,23,26)(16,25,19,20)(18,28,27,21);(2,4)(13,15,17,23)(14,19,22,16)(18,20,27,25)(21,26,28,24);(5,7,9,8)(6,11,10,12)(13,16,17,19)(14,20,22,25)(15,21,23,28)(18,24,27,26);(5,8,9,7)(6,12,10,11)(14,21)(16,24)(18,27)(19,26)(20,25)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250012, #72256979, #72256990, … |
| maximal normal subgroup of | #72250012, #72256979, #72256990, … |
| maximal quotient of | #72264168, #72264385, #72264473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485409);