Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,17,29)(8,12)(9,13,22,28)(10,14)(11,18)(16,30,20,34)(19,25)(21,27,33,36)(23,31)(24,32)(26,35);(1,2,3,4)(8,19)(12,25)(13,28)(15,29)(16,20)(21,33)(23,26)(31,35);(5,7,10,17)(6,11,14,24)(8,20,19,16)(9,18,22,32)(12,26,25,23)(13,15,28,29)(21,35,33,31)(27,34,36,30);(5,8,10,19)(6,12,14,25)(7,16,17,20)(9,21,22,33)(11,23,24,26)(13,27,28,36)(15,30,29,34)(18,31,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312441, #72312448, #72312461, … |
| maximal normal subgroup of | #72312441, #72312448, #72312461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401780);