Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,16,19,9)(5,20)(6,10)(7,11)(8,14)(13,29,31,26)(15,25)(17,23,30,32)(18,27,22,28)(21,24);(1,3)(2,8)(4,17,19,30)(5,21)(6,14)(7,15)(9,26,16,29)(10,25)(11,24)(12,20)(13,18,31,22)(23,28,32,27);(1,4)(2,9)(3,13)(5,18)(6,22)(7,19)(8,23)(10,27)(11,16)(12,26)(14,30)(15,31)(17,21)(20,28)(24,32)(25,29);(1,3)(2,8)(4,13)(5,21)(6,14)(7,15)(9,23)(10,25)(11,24)(12,20)(16,32)(17,18)(19,31)(22,30)(26,28)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241171, #72241172, #72241302, … |
| maximal normal subgroup of | #72241171, #72241172, #72241302, … |
| maximal quotient of | C2×#364608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271804);