Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1496 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)(3,4)(5,6,10,15,11,16,23,33,12,17,9,25,8,14,24,31)(7,18,21,29,19,32,27,30,22,35,20,34,28,36,26,13);(5,7)(6,13,17,30)(8,19)(9,26)(10,27)(11,28)(12,22)(14,29,16,34)(15,32,25,36)(18,31,35,33)(20,24)(21,23);(2,4);(5,8,12,11)(6,14,17,16)(7,19,22,28)(9,23,10,24)(13,29,30,34)(15,31,25,33)(18,32,35,36)(20,26,21,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239524, #72239532, #72239536, … |
| maximal normal subgroup of | #72239524, #72239532, #72239536, … |
| maximal quotient of | #72309905, #72309942, #72310019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401664);