Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,16,19,15,20,14,12)(11,13,21,17,22,24,18,23);(1,2,5,6)(3,7,4,8)(9,11,15,22)(10,17,20,23)(12,13,19,24)(14,18,16,21);(1,3,5,4)(2,7,6,8)(9,12,14,10,15,19,16,20)(11,17,21,13,22,23,18,24);(1,4,5,3)(2,7,6,8)(9,13,16,17,15,24,14,23)(10,18,12,22,20,21,19,11)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72261241 |
| maximal normal subgroup of | #72261241 |
| maximal quotient of | #72262065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348344);