Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15270 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,10,6,11)(3,13,15,24,17,8,16,25)(4,18,20,9,21,28,22,27)(14,30,19,29,31,26,32,23);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,13)(11,25)(12,24)(14,20)(18,23)(21,32)(22,31)(27,30)(28,29);(1,4)(2,9,10,27)(3,14)(5,20)(6,22)(7,21)(8,23,13,29)(11,18,12,28)(15,19)(16,32)(17,31)(24,26,25,30);(8,13)(23,29)(24,25)(26,30)Polycyclic presentation:
2^9|a^2=ef,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216071, #72216130, … |
| maximal normal subgroup of | #72215958, #72216071, #72216130, … |
| maximal quotient of | C2×#419447, #72252114, #72313991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326643);