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 | C2×#3492 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(8,12)(10,11)(14,18)(15,21)(20,23)(22,27);(1,2)(3,4)(5,6,9,7)(8,10,12,11);(6,7)(10,11)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,7,9,6)(8,11,12,10)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27)Polycyclic presentation:
2^9|b^2=i,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311041, #72311050, #72311146, … |
| maximal normal subgroup of | #72311041, #72311050, #72311146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550206);