Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
5546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(13,14)(15,17)(16,19)(18,20);(9,10,11,12)(13,14,17,15)(16,18,20,19);(5,6,8,7)(9,10)(11,12)(14,15)(18,19);(1,2)(3,4)(5,7,8,6)(13,15,17,14)(16,19,20,18);(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^9|b^2=h,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72261333, #72261495, #72261750, … |
| maximal normal subgroup of | #72261333, #72261495, #72261750, … |
| maximal quotient of | #72298413, #72336013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 543629);