Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,13,15,14,16,12,18)(11,17,21,22,19,23,20,24);(5,6,7,8)(10,15)(12,13)(16,18)(17,22)(20,21)(23,24);(1,3,4,2)(10,16)(15,18)(17,23)(22,24);(9,11)(10,17)(12,20)(13,21)(14,19)(15,22)(16,23)(18,24);(11,19)(17,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,f^2=i,g^2=i,b^a=bf,c^a=ci,f^b=fi,g^b=gi,e^d=ei164 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#60176, #72253215, #72269215, … |
| maximal normal subgroup of | C4×#60176, #72284669, #72284953, … |
| maximal quotient of | #72253215, #72269215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277134);