Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| 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×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1137 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)(7,18)(8,16)(9,15)(10,25)(11,27)(12,17)(13,19)(14,23)(20,28)(21,35)(22,33)(24,32)(26,31)(29,34)(30,36);(1,3,4,2)(5,7)(6,13)(8,19)(9,21)(10,26)(11,20)(12,22)(14,28)(15,30)(16,33)(17,31)(18,25)(23,36)(24,34)(27,29)(32,35);(5,8,12,10)(6,14,17,32)(7,19,22,26)(9,23,11,24)(13,28,31,35)(15,25,27,16)(18,29,33,30)(20,34,21,36);(5,9,12,11)(6,15,17,27)(7,20,22,21)(8,23,10,24)(13,29,31,30)(14,25,32,16)(18,28,33,35)(19,34,26,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286581, #72286588, #72286614, … |
| maximal normal subgroup of | #72286581, #72286588, #72286614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401586);