Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,22)(19,21)(23,24);(1,2,3,4);(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284826, #72285076 |
| maximal normal subgroup of | #72284826, #72285076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278034);