Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,17,15,11,13,19)(12,20,24,18,22,21,23,16);(1,2)(3,4)(5,7,8,6)(9,11,15,10)(12,21,22,20)(13,17,14,19)(16,23,18,24);(2,4)(6,7)(9,12,13,23,15,22,14,24)(10,16,17,20,11,18,19,21);(12,22)(16,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238814, #72239940 |
| maximal normal subgroup of | #72238814, #72239940 |
| maximal quotient of | #72238820, #72239947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265558);