Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500005 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23960 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
907 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)(3,13)(4,10)(5,19)(6,24)(7,12)(8,16)(9,21)(11,20)(14,28)(15,31)(17,32)(18,30)(22,29)(23,27)(25,26);(3,14,15,18)(4,5)(6,20)(8,26,27,32)(9,29)(10,19)(11,24)(13,30,31,28)(16,17,23,25)(21,22);(1,3)(2,8)(4,18)(5,23)(6,25)(7,15)(9,26)(10,13)(11,30)(12,27)(14,22)(16,21)(17,20)(19,32)(24,28)(29,31);(1,2)(3,8)(4,19)(5,10)(6,11)(7,12)(9,22)(13,23)(14,26)(15,27)(16,31)(17,28)(18,32)(20,24)(21,29)(25,30)Polycyclic presentation:
2^9|b^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231708, #72231712, … |
| maximal normal subgroup of | #72213945, #72231708, #72231712, … |
| maximal quotient of | #72226819, #72239881, #72250626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407201);