Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15267 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,22,9)(3,12,15,7,16,14,24,25)(4,17,19,27,21,26,20,8)(13,29,31,30,18,23,32,28);(1,3,5,15,6,16,22,24)(2,7,9,12,11,25,10,14)(4,18,19,32,21,13,20,31)(8,23,26,30,27,29,17,28);(1,4)(2,8)(3,13)(5,20)(6,21)(7,23)(9,17)(10,26)(11,27)(12,28)(14,30)(15,32)(16,18)(19,22)(24,31)(25,29);(1,2)(3,14)(4,8)(5,10)(6,11)(7,24)(9,22)(12,16)(13,30)(15,25)(17,19)(18,28)(20,26)(21,27)(23,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72219688, #72219690, … |
| maximal normal subgroup of | #72215974, #72219688, #72219690, … |
| maximal quotient of | C2×#419045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326241);