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 | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(7,11)(10,12)(14,15)(16,23)(20,22)(24,28);(1,2)(3,4)(6,9)(10,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72303257, #72342339, #72342343, … |
| maximal normal subgroup of | #72303257, #72342339, #72342343, … |
| maximal quotient of | #72303275, #72303278, #72303306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549061);