Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
22517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8);(5,6)(7,8)(9,10,15,19)(11,18,14,16)(12,13,17,24)(20,23,22,21);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,21,24,23);(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24);(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,22)(18,20);(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,23,24,21)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72249616, #72256860, #72262355, … |
| maximal normal subgroup of | #72249616, #72256860, #72262355, … |
| maximal quotient of | #72264057, D4×#2409, #72264083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540429);