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 | |
11036 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)(12,20)(16,21)(19,23)(22,24);(5,6)(7,8)(10,14)(11,17)(16,21)(19,23);(2,3)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22);(1,3,4,2);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72250400, #72257350, #72257360, … |
| maximal normal subgroup of | #72250400, #72257350, #72257360, … |
| maximal quotient of | #72257188, #72257914, #72280291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585155);