Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1193 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)(16,27)(17,28)(19,24)(20,25)(21,26)(22,30)(23,31)(29,35)(32,34)(33,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,18)(10,16)(12,23)(13,15)(14,22)(19,31)(20,29)(21,30)(24,28)(25,33)(26,27)(32,36)(34,35);(7,16)(9,21)(11,22)(15,27)(17,29)(19,32)(23,33)(28,35);(5,8,10,20)(6,12,14,25)(7,17,16,29)(9,19,21,32)(11,23,22,33)(13,24,26,34)(15,28,27,35)(18,31,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,b^a=be,c^a=cf,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312325, #72312334, #72312352, … |
| maximal normal subgroup of | #72312325, #72312334, #72312352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401544);