Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481601 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,15,22,16)(13,17)(14,21,23,19)(20,24);(1,2,4,6)(3,8,7,5)(9,11)(10,14,17,23)(12,19)(13,20)(15,18,16,24)(21,22);(9,12)(10,15)(11,19)(13,22)(14,18)(16,17)(20,21)(23,24);(11,20)(14,23)(18,24)(19,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=hi,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282315, #72282336, #72282374, … |
| maximal normal subgroup of | #72282315, #72282336, #72282374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389313);