Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,17,13,15,23,18)(11,19,21,16,22,14,20,24);(10,14)(11,20)(12,23)(15,19)(16,17)(18,24)(21,22);(2,5)(6,8)(10,15)(14,19)(16,24)(17,18);(10,15)(14,19)(16,24)(17,18)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#481279, #72213530, #72216543, … |
| maximal normal subgroup of | C2×#481279, #72213530, #72216543, … |
| maximal quotient of | C2×#481279, #72285967, #72286089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388475);