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 | |
10926 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22);(5,6,7,8)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,21,24,23);(2,3)(10,17)(11,12)(14,24)(20,22);(1,3,4,2)(9,12,15,11)(10,18,17,16)(13,20,19,22)(14,23,24,21);(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,23,18,21);(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,20)(18,22)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,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 | #72234485, #72234488, #72239419, … |
| maximal normal subgroup of | #72234485, #72234488, #72239419, … |
| maximal quotient of | #72261776, #72280828, #72280883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590978);