Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,20)(15,16,19,17);(1,2)(3,4)(13,15,17,14,18,19,16,20);(2,4)(6,9)(10,12)(13,16,18,17)(14,15,20,19);(2,4)(13,16,18,17)(14,19,20,15);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,19,20,15)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214806, #72263947, #72263954, … |
| maximal normal subgroup of | #72214806, #72263947, #72263954, … |
| maximal quotient of | #72214556 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361768);