Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500355 |
| 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 | #61267 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2930 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(4,16)(5,17)(6,20)(9,11)(10,24)(12,18)(13,27)(14,30)(19,28)(21,22)(23,25)(26,29)(31,32);(2,9,11,8)(4,17)(5,16)(6,20)(10,24,25,23)(12,21,22,18)(13,28)(14,30)(19,27)(26,29,31,32);(1,2,6,22,7,11,20,12)(3,10,14,31,15,25,30,26)(4,18,16,9,5,21,17,8)(13,29,27,23,19,32,28,24);(1,3)(2,10)(4,19)(5,13)(6,14)(7,15)(8,23)(9,24)(11,25)(12,26)(16,28)(17,27)(18,32)(20,30)(21,29)(22,31)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,g^2=hi,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=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231596, #72250762, … |
| maximal normal subgroup of | #72213945, #72231596, #72250762, … |
| maximal quotient of | #72224931, #72237227, #72250661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407551);