Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3714 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,22,24,26)(23,27,28,25);(5,6,8,10)(7,9,12,11)(13,15,17,20)(14,16,18,19);(7,12)(9,11)(13,14,17,18)(15,19,20,16)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,16)(17,18)(19,20);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281048, #72281545, #72281913, … |
| maximal normal subgroup of | #72281048, #72281545, #72281913, … |
| maximal quotient of | #72328384 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 991063);