Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1083 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)(3,11)(4,16,17,8)(5,9)(6,10)(7,13,21,30)(12,25)(14,26)(15,27)(18,31,32,23)(19,24)(20,28,22,29);(1,3,6,15)(2,7,10,21)(4,13,17,30)(5,12,19,14)(8,11,16,27)(9,20,24,22)(18,28,32,29)(23,25,31,26);(3,12)(4,17)(5,19)(7,20)(9,24)(11,25)(13,29)(14,15)(21,22)(23,31)(26,27)(28,30);(2,8)(7,11)(9,23)(10,16)(20,25)(21,27)(22,26)(24,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=ehi,g^c=gi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237521, #72237525, #72237549, … |
| maximal normal subgroup of | #72237521, #72237525, #72237549, … |
| maximal quotient of | #72312408, #72312536, #72312591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408388);