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 | C23×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1228 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)(7,11)(8,18)(9,21)(10,14)(12,20)(13,23)(15,30)(16,25)(17,28)(19,29)(22,27)(24,32)(26,33)(31,35)(34,36);(1,3,4,2)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,28)(14,25)(18,35)(19,31)(20,32)(21,26)(22,34)(23,33)(27,36)(29,30);(5,8,10,19)(6,12,14,27)(7,15,16,31)(9,22,23,20)(11,24,25,36)(13,29,21,18)(17,34,33,32)(26,35,28,30);(7,16)(11,25)(15,31)(17,33)(24,36)(26,28)(30,35)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286077, #72286120, #72286145, … |
| maximal normal subgroup of | #72286077, #72286120, #72286145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401567);