Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,15,13)(11,16,12,17)(14,21,22,18)(19,24,20,23);(5,6,7,8)(9,11)(10,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24);(2,3)(10,13)(11,12)(18,21)(19,20);(1,3,4,2)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311237, #72311254, #72311295, … |
| maximal normal subgroup of | #72311237, #72311254, #72311295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581800);