Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3455 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)(5,6,9,7)(8,11,12,10)(13,14,17,16)(15,18,20,19)(22,23)(24,28);(5,6,9,7)(8,11,12,10)(13,15,17,20)(14,18,16,19)(21,22,25,23)(24,27,28,26);(6,7)(10,11)(14,16)(18,19)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,17,14)(15,19,20,18)(21,23,25,22)(24,26,28,27);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72332006, #72332011, #72332054 |
| maximal normal subgroup of | #72332006, #72332011, #72332054 |
| maximal quotient of | #72333069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5513288);