Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,13,18,21)(12,15,20,22)(14,16)(17,23)(19,24);(5,6)(7,8)(9,11)(10,13)(12,17)(14,18)(15,19)(16,21)(20,24)(22,23);(2,4)(6,8)(9,12,14,20)(10,15,16,22)(11,17,18,24)(13,19,21,23);(9,13)(10,11)(12,19)(14,21)(15,17)(16,18)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=ef,c^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240661, #72240667, #72240675, … |
| maximal normal subgroup of | #72240661, #72240667, #72240675, … |
| maximal quotient of | #72249506, #72249520, #72317095 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302615);