Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460969 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16375 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
960 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)(3,11,14,7)(4,15)(5,19)(6,10)(8,18)(9,20)(12,21,16,23)(13,22,28,26)(17,25)(24,30)(27,31,29,32);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,23,15,21)(9,22)(10,11)(17,29,30,27)(19,26)(20,28)(24,32,25,31);(1,4)(2,8,10,15)(3,12)(5,17)(6,18)(7,21,11,23)(9,25,19,24)(13,27)(14,16)(20,30)(22,32,26,31)(28,29);(1,5)(2,9,10,19)(3,13)(4,17)(6,20)(7,22,11,26)(8,24,15,25)(12,27)(14,28)(16,29)(18,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237789, #72237790, #72237796, … |
| maximal normal subgroup of | #72237789, #72237790, #72237796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368165);