Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342574 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(10,14)(12,19)(15,20)(18,23);(1,3,5,4)(2,6,7,8)(10,14)(15,20)(16,21)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=g,c^2=hi,g^2=i,b^a=bf,c^a=ci,f^a=fh,g^b=gh,e^d=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281576, #72282071 |
| maximal normal subgroup of | #72281576, #72282071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249777);