Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2617 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(11,15)(14,16)(19,23)(21,24);(1,2,4,3)(10,13)(14,16)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(17,19)(18,21)(20,23)(22,24);(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|b^2=i,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72266737, #72336090, #72336093, … |
| maximal normal subgroup of | #72266737, #72336090, #72336093, … |
| maximal quotient of | #72267000, #72336202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215328);