Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35237 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,31,32,30);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(11,16)(13,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(18,24)(19,27)(23,25)(28,32);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fhi,e^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=efhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72316766, #72316936, #72316949, … |
| maximal normal subgroup of | #72316766, #72316936, #72316949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6772897);