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 | |
11352 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)(9,10,14,16)(11,15,19,23)(12,17,21,24)(13,18,20,22);(5,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,22)(18,24);(2,3)(10,16)(11,19)(12,21)(18,22);(1,3,4,2)(9,12,14,21)(10,17,16,24)(11,20,19,13)(15,22,23,18);(9,13,14,20)(10,18,16,22)(11,12,19,21)(15,17,23,24);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72239419, #72302972, #72302976, … |
| maximal normal subgroup of | #72239419, #72302972, #72302976, … |
| maximal quotient of | #72280852, #72280871, #72280903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590063);