Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| 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 | C27 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
10066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,17)(15,19,20,18)(21,22,25,27)(23,24,28,26);(6,10)(7,12)(14,17)(18,19)(21,23,25,28)(22,26,27,24);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20);(5,8)(6,7)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72241065, #72241214, #72285954, … |
| maximal normal subgroup of | #72241065, #72241214, #72285954, … |
| maximal quotient of | #72330191, #72331287, #72335972 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752902);