Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,15,10,13,23,28,11,14,24,29,8,21,22,34)(7,16,19,31,20,26,33,35,18,30,32,36,17,12,25,27);(1,2,3,4)(5,7,8,17,11,18,10,20)(6,12,13,16,14,26,21,30)(9,19,22,25,24,32,23,33)(15,27,28,31,29,35,34,36);(12,26)(16,30)(27,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239415, #72239416, #72239426, … |
| maximal normal subgroup of | #72239415, #72239416, #72239426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55173);