Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500349 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24089 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61293 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,13)(4,18)(5,21)(6,20)(8,24)(9,10)(11,22)(14,30)(15,31)(16,17)(19,29)(23,32)(25,28)(26,27);(1,2)(3,14,15,27)(4,10)(5,19)(6,22)(7,12)(8,16,25,32)(9,18)(11,20)(13,26,31,30)(17,24,23,28)(21,29);(1,3,6,23,7,15,20,17)(2,8,11,30,12,25,22,26)(4,16,18,31,21,32,5,13)(9,27,10,28,19,14,29,24);(1,2)(3,8)(4,19)(5,10)(6,11)(7,12)(9,21)(13,28)(14,16)(15,25)(17,26)(18,29)(20,22)(23,30)(24,31)(27,32)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231593, #72231596, … |
| maximal normal subgroup of | #72213937, #72231593, #72231596, … |
| maximal quotient of | #72231555, #72237229, C2×#500349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407545);