Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,13,17,14,12,16,21)(11,15,20,23,18,19,22,24);(1,2,3,4)(9,11)(10,15)(12,19)(13,20)(14,18)(16,22)(17,23)(21,24);(5,6,7,8)(11,18)(15,19)(20,22)(23,24);(9,12,13,21,14,10,16,17)(11,19,20,24,18,15,22,23);(10,12)(15,19)(17,21)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72257303, #72257519, #72261157, … |
| maximal normal subgroup of | #72257303, #72257519, #72261157, … |
| maximal quotient of | #72257825, #72257842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280993);