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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,10,8,12)(13,14,17,19)(15,18,20,16)(21,22,25,26)(23,24,28,27);(5,7,9,8)(6,10,11,12)(13,15,17,20)(14,18,19,16)(22,26)(24,27);(6,11)(10,12)(14,19)(15,20)(21,23,25,28)(22,24,26,27);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,20)(17,19)(21,23)(22,27)(24,26)(25,28);(5,8,9,7)(6,12,11,10)(13,16,17,18)(14,20,19,15)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,e^2=fg,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 | #72329366, #72329371, #72329434 |
| maximal normal subgroup of | #72329366, #72329371, #72329434 |
| maximal quotient of | #72331392 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4839983);