Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442700 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2408 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,18)(5,22)(6,10)(7,11)(8,15)(9,21)(13,24)(14,17)(16,25)(19,26)(20,28)(23,27)(29,31)(30,32);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,14)(11,25)(12,22)(13,21)(18,32)(20,31)(23,30)(24,28)(27,29);(1,4,5,21,7,20,6,23)(2,9,10,18,11,27,22,28)(3,13,16,31,17,30,15,19)(8,24,14,26,25,32,12,29);(1,2)(3,14)(4,9)(5,10)(6,22)(7,11)(8,15)(12,17)(13,26)(16,25)(18,21)(19,24)(20,27)(23,28)(29,30)(31,32);(4,20)(9,27)(13,30)(18,28)(19,31)(21,23)(24,32)(26,29)Polycyclic presentation:
2^9|c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72213913, #72214632, … |
| maximal normal subgroup of | #72213603, #72213913, #72214632, … |
| maximal quotient of | C2×#6442700, #72255594, #72255864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349896);