Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419907 |
| 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 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,15,8,16,24,22,25)(4,17,20,28,21,9,19,27)(14,26,18,23,29,30,32,31);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,26)(10,24)(11,13)(12,25)(14,19)(17,30)(20,29)(21,32)(23,27)(28,31);(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,22)(14,26)(15,25)(16,24)(17,21)(18,31)(19,27)(20,28)(23,32)(29,30);(1,4,5,19,7,21,6,20)(2,9,10,27,12,17,11,28)(3,14,15,18,16,29,22,32)(8,23,13,26,25,31,24,30)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72216144, #72219691, … |
| maximal normal subgroup of | #72215981, #72216144, #72219691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327103);