Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36614 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,20,24)(15,16,18,21)(17,22,28,26)(19,27,25,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,18)(14,21,24,16)(17,25,28,19)(22,27,26,23);(6,9)(10,12)(13,16,20,21)(14,15,24,18)(17,27,28,23)(19,26,25,22);(13,17,20,28)(14,22,24,26)(15,25,18,19)(16,23,21,27);(5,6)(7,10)(8,9)(11,12)(13,18,20,15)(14,16,24,21)(17,19,28,25)(22,23,26,27);(5,7)(6,10)(8,11)(9,12)(13,19)(14,23)(15,17)(16,26)(18,28)(20,25)(21,22)(24,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304569, #72304580, #72304636, … |
| maximal normal subgroup of | #72304569, #72304580, #72304636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615985);