Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442595 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2404 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3843 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)(4,9)(5,20)(6,11)(7,10)(8,14)(13,29)(15,23)(16,24)(17,22)(18,28)(19,26)(21,27)(25,31)(30,32);(1,3)(2,8)(4,17)(5,16)(6,14)(7,15)(9,25)(10,24)(11,23)(12,20)(13,18)(19,30)(21,31)(22,28)(26,29)(27,32);(1,4)(2,9)(3,13,15,31)(5,21)(6,18)(7,19)(8,22,24,32)(10,26)(11,28)(12,25,23,29)(14,30,16,17)(20,27);(3,14,15,16)(8,23,24,12)(13,30,31,17)(22,29,32,25);(2,10)(8,24)(9,26)(11,20)(12,23)(22,32)(25,29)(27,28)Polycyclic presentation:
2^9|c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249649, #72249829, #72285925, … |
| maximal normal subgroup of | #72249649, #72249829, #72285925, … |
| maximal quotient of | #72256028, #72257172, #72302433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358971);