Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493635 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63020 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,16,26)(3,8,12,22,13,23,17,24)(4,7,15,21,14,20,18,25)(11,19,28,31,27,30,29,32);(2,7)(5,16)(8,19)(9,21)(10,20)(12,17)(15,18)(22,31)(23,30)(24,32)(25,26)(28,29);(4,14)(11,27)(15,18)(28,29);(1,3,6,13)(2,8,10,23)(4,11,14,27)(5,17,16,12)(7,19,20,30)(9,24,26,22)(15,29,18,28)(21,32,25,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72303256 |
| maximal normal subgroup of | #72303256 |
| maximal quotient of | #72303269, #72311121 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400840);