Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5715 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)(5,6,7,8)(13,14,18,19)(15,17,16,20);(5,6,7,8)(10,12)(14,19)(15,16);(5,6)(7,8)(9,10)(11,12)(13,15,18,16)(14,20,19,17);(1,2)(3,4)(5,6)(7,8)(13,16,18,15)(14,17,19,20);(13,17)(14,15)(16,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,d^2=f,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 | #72263592, #72263608, #72263616, … |
| maximal normal subgroup of | #72263592, #72263608, #72263616, … |
| maximal quotient of | #72298234, #72298276, C2×#845348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752544);