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) = C24×C43 |
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 | |
4164 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)(8,10,12,11)(14,16)(15,19)(21,22)(23,27)(24,25)(26,28);(8,12)(10,11)(13,14,17,16)(15,18,19,20)(21,23,25,28)(22,26,24,27);(6,7)(10,11)(15,19)(18,20)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,18,19,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=h,d^2=g,e^2=fg,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 | #72232524, #72232534, #72232545, … |
| maximal normal subgroup of | #72232524, #72232534, #72232545, … |
| maximal quotient of | #72328472, #72328961, #72329087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5518869);