Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C2×#2669 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(14,17)(15,19)(22,23)(24,28);(2,4)(5,7,9,8)(6,10,11,12)(13,14,16,17)(15,20,19,18)(22,23)(26,27);(1,2)(3,4)(6,11)(10,12)(14,17)(18,20)(21,22,25,23)(24,26,28,27);(5,8,9,7)(6,12,11,10)(13,15,16,19)(14,18,17,20)(21,23,25,22)(24,27,28,26);(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,17)(18,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72332191, #72332220, #72332267, … |
| maximal normal subgroup of | #72332191, #72332220, #72332267, … |
| maximal quotient of | #72332973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3902415);