Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,20,33)(8,13,22,32)(9,14)(10,15)(11,18)(12,21,30,24)(17,28)(19,26,23,29)(25,31)(27,35)(34,36);(1,2,3,4)(5,7)(6,11)(8,18)(9,23)(10,20)(12,29)(13,16)(14,27)(15,28)(17,22)(19,25)(21,34)(24,35)(26,30)(31,36)(32,33);(5,8)(6,12)(7,17)(9,24)(10,22)(11,26)(13,31)(14,32)(15,30)(16,27)(18,20)(19,35)(21,25)(23,34)(28,29)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=ci,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284568, #72284597, #72284631, … |
| maximal normal subgroup of | #72284568, #72284597, #72284631, … |
| maximal quotient of | #72303936, #72303966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53832);