Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,22)(13,17)(14,18)(15,20)(16,23)(21,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,13)(14,21)(16,24)(17,19)(18,22)(20,23);(2,6)(5,7)(9,10)(11,19)(12,15)(13,16)(14,18)(17,23)(20,22)(21,24);(3,8)(5,7)(9,12,23,21,14,20,13,11)(10,15,17,24,18,22,16,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339040, #72339041, #72339042 |
| maximal normal subgroup of | #72339040, #72339041, #72339042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337531);