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 | #24364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2073 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)(5,6,8,9)(7,10,11,12)(14,20)(15,23)(21,25)(22,27);(6,9)(10,12)(13,14,18,20)(15,24,23,28)(16,21,26,25)(17,22,19,27);(2,4)(13,15,18,23)(14,16,20,26)(17,25,19,21)(22,28,27,24);(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,21,20,25)(15,22,23,27)(17,24,19,28);(13,17)(14,20)(15,21)(18,19)(23,25)(24,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256952, #72256990, #72257004 |
| maximal normal subgroup of | #72256952, #72256990, #72257004 |
| maximal quotient of | #72264155, #72264909, #72265166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483362);