Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,18,15,12,14,17)(11,19,23,21,24,20,22,16);(5,6)(7,8)(9,11,13,23,15,24,14,22)(10,16,18,19,12,21,17,20);(1,2,3,4)(6,8)(9,12)(10,15)(11,20)(13,18)(14,17)(16,22)(19,24)(21,23);(9,10)(11,21)(12,15)(13,18)(14,17)(16,24)(19,22)(20,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216146 |
| maximal normal subgroup of | #72216146 |
| maximal quotient of | #72215990 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276046);