Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
5278 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)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24);(10,13)(14,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,4)(5,6)(7,8)(11,16)(14,15)(17,19,20,24)(18,22,23,21);(5,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72219266, #72260456, #72260775, … |
| maximal normal subgroup of | #72219259, #72219265, #72219266, … |
| maximal quotient of | #72323507, #72323623, #72330477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 996115);