Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30396 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,9)(3,15)(4,18)(5,7)(6,23)(10,26)(11,30)(12,24)(13,14)(16,29)(17,31)(19,25)(20,21)(22,27);(1,2)(3,16)(4,19)(5,13)(6,24)(7,9)(8,14)(10,17)(11,20)(12,27)(15,18)(21,25)(22,28)(23,32)(26,30)(29,31);(1,3)(2,10)(4,14)(5,17)(6,25)(7,19)(8,11)(9,16)(12,18)(13,20)(15,23)(21,32)(22,30)(24,26)(27,31)(28,29);(1,4)(2,11)(3,14)(5,16)(6,26)(7,20)(8,10)(9,17)(12,31)(13,19)(15,32)(18,27)(21,23)(22,29)(24,25)(28,30)Polycyclic presentation:
2^9|e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^d=fh,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#509107, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#509107, #72320520, … |
| maximal quotient of | C2×#509107, #72250029, #72250036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416303);