Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5442 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)(11,16)(12,14)(13,15)(17,18,21,22)(19,24,20,23);(9,11)(10,12)(13,14)(15,16)(18,22)(23,24);(2,3)(11,14)(12,16)(17,19,21,20)(18,23,22,24);(1,3,4,2)(5,6)(7,8)(9,12,13,16)(10,14,15,11)(17,20,21,19)(18,24,22,23);(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=fghi,c^2=fh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219267, #72297488, #72297533, … |
| maximal normal subgroup of | #72219265, #72219267, #72297488, … |
| maximal quotient of | #72298109, #72330302, #72330492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851191);