Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×B(4,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,28)(17,29)(19,25)(20,26)(21,34)(22,27)(23,30)(24,32)(31,36)(33,35);(5,7,10,17)(6,11,14,24)(8,16,19,31)(9,18,22,32)(12,23,25,35)(13,15,27,29)(20,30,21,33)(26,28,34,36);(5,8)(6,12)(7,16)(9,21)(10,19)(11,23)(13,26)(14,25)(15,28)(17,31)(18,33)(20,22)(24,35)(27,34)(29,36)(30,32);(1,3,4,2)(7,17)(8,19)(11,24)(12,25)(15,29)(18,32)(20,21)(26,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312322, #72312330, #72312334, … |
| maximal normal subgroup of | #72312322, #72312330, #72312334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402941);