Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,17,11,16,10,15)(7,18,21,12,23,30,22,29)(8,14,25,32,26,13,24,31)(19,34,20,27,36,28,35,33);(1,2)(3,4)(5,7,10,22,11,23,9,21)(6,12,17,30,16,29,15,18)(8,19,25,20,26,36,24,35)(13,27,32,34,14,33,31,28);(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,27)(14,16)(15,31)(17,32)(18,28)(20,22)(21,35)(23,36)(29,33)(30,34);(2,4)(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,28)(13,16)(15,31)(17,32)(18,27)(19,22)(21,36)(23,35)(29,34)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311927, #72311981, #72312189, … |
| maximal normal subgroup of | #72311927, #72311981, #72312189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263303);