Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5094 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)(6,8)(9,10)(11,15)(12,13)(14,16)(18,22)(20,24);(2,4)(5,6,7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,20)(21,22)(23,24);(2,4)(5,6,7,8)(10,13)(14,15)(18,22)(19,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24);(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|b^2=fghi,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232127, #72232131, #72232185, … |
| maximal normal subgroup of | #72232127, #72232131, #72232185, … |
| maximal quotient of | #72278594, #72279181, #72299543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363498);