Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| 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 | |
2376 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)(4,15)(5,10)(6,20)(7,12)(8,14)(9,13)(16,29)(17,27)(18,24)(19,23)(21,26)(22,30)(25,28)(31,32);(1,3,7,6)(2,8,12,23)(4,13,17,21)(5,14,18,19)(9,22,26,29)(10,11,24,20)(15,25,27,32)(16,28,30,31);(5,18)(14,19)(16,30)(28,31);(1,4,7,17)(2,9,12,26)(3,13,6,21)(5,16,18,30)(8,22,23,29)(10,25,24,32)(11,27,20,15)(14,28,19,31)Polycyclic presentation:
2^9|b^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,f^b=fhi,e^c=eh,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237521, #72237540, #72260481, … |
| maximal normal subgroup of | #72237521, #72237540, #72260481, … |
| maximal quotient of | #72261102, #72261171, #72281281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419018);