Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2099 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)(7,11)(8,12)(9,10)(15,21)(16,24)(18,25)(20,26)(22,27)(23,28);(1,2)(3,4)(6,10)(11,12)(13,14,17,19)(15,22,23,18)(16,20,24,26)(21,27,28,25);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,21,17,23,24,28)(14,18,20,25,19,22,26,27);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,23)(18,22)(21,28)(25,27);(2,4)(14,19)(18,22)(20,26)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257534, #72257619, #72263869 |
| maximal normal subgroup of | #72257534, #72257619, #72263869 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491852);