Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451734 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3933 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)(3,11)(4,8)(5,18)(6,9)(7,13)(10,21)(12,30)(14,27)(15,29)(16,28)(17,25)(19,23)(20,26)(22,31)(24,32);(1,3)(2,7)(4,15)(5,19)(6,14)(8,24)(9,23)(10,27)(11,18)(12,17)(13,21)(16,31)(20,32)(22,25)(26,30)(28,29);(1,4)(2,8)(3,12,14,15)(5,20)(6,17)(7,22,23,24)(9,25)(10,28)(11,29,27,30)(13,32,19,31)(16,21)(18,26);(1,4)(2,8)(3,12,14,15)(5,16)(6,17)(7,22,23,24)(9,25)(10,26)(11,30,27,29)(13,31,19,32)(18,28)(20,21);(2,9)(7,23)(8,25)(10,18)(11,27)(22,24)(26,28)(29,30)Polycyclic presentation:
2^9|c^2=g,d^2=g,f^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,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72234100, #72240647, #72249649, … |
| maximal normal subgroup of | #72234100, #72240647, #72249649, … |
| maximal quotient of | #72233837, #72233855, #72256098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358930);