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) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3676 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)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,24,20,23);(6,8)(10,12)(14,15)(18,22)(19,20);(5,6)(7,8)(17,19,21,20)(18,23,22,24);(1,2,3,4)(9,11,13,16)(10,14,12,15)(17,20,21,19)(18,23,22,24);(9,12,13,10)(11,14,16,15)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=gi,e^2=fg,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 | #72323405, #72323407, #72323417, … |
| maximal normal subgroup of | #72323405, #72323407, #72323417, … |
| maximal quotient of | #72323679, C2×#2921768 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2828964);