Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17155 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
715 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)(4,8,17,23)(5,9,18,25)(6,10)(12,21)(13,26)(15,19)(16,24)(20,28)(22,30)(27,31,29,32);(1,3)(2,7)(4,15,17,12)(5,13)(6,14)(8,21,23,19)(9,20)(10,11)(16,27,30,29)(18,28)(22,31,24,32)(25,26);(1,4)(2,8,10,23)(3,12)(5,16)(6,17)(7,19,11,21)(9,24,25,22)(13,27)(14,15)(18,30)(20,32,26,31)(28,29);(1,5)(2,9,10,25)(3,13)(4,16)(6,18)(7,20,11,26)(8,22,23,24)(12,27)(14,28)(15,29)(17,30)(19,31,21,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237954, #72237958, #72237962, … |
| maximal normal subgroup of | #72237954, #72237958, #72237962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385777);