Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
708 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)(4,8)(5,20)(6,10)(7,13)(9,17)(11,18)(14,21)(15,24)(16,22)(19,25)(23,31)(26,30)(27,32)(28,29);(1,3,5,15,6,13,18,29)(2,7,11,24,10,12,20,28)(4,16,17,31,19,14,30,32)(8,21,26,23,25,22,9,27);(2,8)(3,13)(4,17)(5,18)(7,21)(9,11)(10,25)(12,22)(14,31)(16,32)(19,30)(20,26)(23,28)(24,27);(2,8)(3,13)(4,17)(5,18)(7,22)(9,11)(10,25)(12,21)(14,31)(16,32)(19,30)(20,26)(23,24)(27,28)Polycyclic presentation:
2^9|b^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,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,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#509113, #72320552, … |
| maximal normal subgroup of | #72237280, C2×#509113, #72320552, … |
| maximal quotient of | C2×#509113, #72250953, #72257138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416309);