Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| 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 | |
3829 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)(4,5)(6,10)(7,24)(8,11)(9,15)(13,31)(14,30)(16,27)(17,29)(18,21)(19,23)(20,32)(22,26)(25,28);(1,3)(2,9)(4,17)(5,20)(6,15)(7,25)(8,16)(10,27)(11,28)(12,24)(13,19)(14,23)(18,30)(21,29)(22,31)(26,32);(1,4)(2,5)(3,13,16,17)(6,18)(7,26)(8,19)(9,14,28,20)(10,21)(11,23)(12,29,27,31)(15,32,25,30)(22,24);(1,5)(2,4)(3,14,16,20)(6,21)(7,22)(8,23)(9,13,28,17)(10,18)(11,19)(12,30,27,32)(15,31,25,29)(24,26);(1,6,8,7)(2,10,11,24)(3,15,16,25)(4,18,19,26)(5,21,23,22)(9,27,28,12)(13,32,17,30)(14,31,20,29)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261089, #72261154, #72263053, … |
| maximal normal subgroup of | #72261089, #72261154, #72263053, … |
| maximal quotient of | #72262969, #72284254, #72285557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358929);