Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1607 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)(4,9,20,27)(5,22)(6,10)(7,11)(8,15)(13,17)(14,28,29,25)(16,24)(18,31,30,23)(19,32,21,26);(1,3)(2,8)(4,18)(5,16)(6,15)(7,17)(9,25)(10,13)(11,24)(12,22)(14,19)(20,30)(21,29)(23,26)(27,28)(31,32);(1,2)(3,13)(4,9)(5,10)(6,22)(7,11)(8,15)(12,17)(14,28)(16,24)(18,23)(19,26)(20,27)(21,32)(25,29)(30,31);(1,4,5,19,7,20,6,21)(2,9,10,26,11,27,22,32)(3,14,15,18,17,29,16,30)(8,23,12,25,24,31,13,28)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215938, #72215974, #72215981, … |
| maximal normal subgroup of | #72215938, #72215974, #72215981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327095);