Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25508 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32);(9,11)(10,13)(12,16)(14,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,30,31,32);(1,3,5,4)(2,6,8,7)(11,16)(13,14)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,28,29,32);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,16)(13,15)(18,24)(19,25)(22,29)(26,31);(1,4,5,3)(2,6,8,7)(11,16)(13,14)(18,24)(22,29)(23,30)(28,32)Polycyclic presentation:
2^9|b^2=fh,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257531, #72257553, #72264263, … |
| maximal normal subgroup of | #72257531, #72257553, #72264263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6523914);