Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511825 |
| 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 | |
2396 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)(3,13)(4,10)(5,19)(6,12)(7,23)(8,14)(9,16)(11,15)(17,29)(18,25)(20,32)(21,27)(22,26)(24,30)(28,31);(1,3,8,7)(2,9,14,26)(4,17,5,15)(6,16,21,22)(10,28,19,24)(11,25,29,32)(12,13,27,23)(18,31,20,30);(1,4,8,5)(2,10,14,19)(3,15,7,17)(6,20,21,18)(9,24,26,28)(11,23,29,13)(12,25,27,32)(16,31,22,30);(1,5,8,4)(2,11,14,29)(3,15,7,17)(6,20,21,18)(9,25,26,32)(10,23,19,13)(12,24,27,28)(16,30,22,31)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260417, #72260481, C2×#511830, … |
| maximal normal subgroup of | #72260417, #72260481, C2×#511830, … |
| maximal quotient of | #72260501, #72261129, #72261292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419026);