Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,23)(22,31)(24,27)(26,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,20)(26,31)(27,29);(11,16)(13,15)(17,19,21,20)(18,24,23,32)(22,29,30,27)(25,26,28,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,19)(18,25,23,28)(22,29,30,27)(24,26,32,31);(2,5)(6,8)(19,20)(24,32)(25,28)(27,29)Polycyclic presentation:
2^9|c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72257512, #72257558, #72264310 |
| maximal normal subgroup of | #72257512, #72257558, #72264310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678118);