Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,15,23,29,11,8,13,20,24,21,27,33)(7,16,18,28,19,30,32,36,17,12,22,25,31,26,34,35);(1,2,3,4)(5,7)(6,12)(8,16)(9,22)(10,19)(11,17)(13,18)(14,28)(15,26)(20,25)(21,30)(23,34)(24,31)(27,32)(29,36)(33,35);(5,8,9,20,10,21,23,33,11,6,13,14,24,15,27,29)(7,16,22,25,19,30,34,35,17,12,18,28,31,26,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^b=ei,f^b=fi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239200, #72239201, #72239414, … |
| maximal normal subgroup of | #72239200, #72239201, #72239414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54556);