Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494058 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20739 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#576 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
945 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)(3,12,14,7)(4,8)(5,19)(6,9)(10,16)(11,17)(13,25,30,20)(15,22,29,27)(18,23)(21,28,26,32)(24,31);(1,3)(2,7)(4,13)(5,15)(6,14)(8,20)(9,12)(10,21)(11,22)(16,28)(17,29)(18,30)(19,27)(23,25)(24,26)(31,32);(2,8)(4,16)(5,17)(7,20)(9,23)(10,11)(12,25)(13,28)(15,29)(18,31)(19,24)(21,22)(26,27)(30,32);(1,4,5,16,6,18,17,31)(2,8,19,10,9,23,11,24)(3,13,15,28,14,30,29,32)(7,20,27,21,12,25,22,26)Polycyclic presentation:
2^9|a^2=g,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,g^b=gi,d^c=dh,f^c=fh,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72213945, #72231704, … |
| maximal normal subgroup of | #72213937, #72213945, #72231704, … |
| maximal quotient of | #72224839, #72231520, C2×#494058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401254);