Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2669 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,14)(12,16)(13,15)(19,23)(21,24);(2,4)(6,8)(9,11,13,16)(10,14,15,12);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,20,22)(19,21,23,24);(2,4)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72330820, #72330870, #72330902, … |
| maximal normal subgroup of | #72330820, #72330870, #72330902, … |
| maximal quotient of | #72332782 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2762733);