Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 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)(3,7)(4,6)(5,8)(17,18)(19,27)(20,26)(21,25)(22,31)(23,28)(24,29)(30,32);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,14,12,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,27)(31,32);(3,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,26)(25,32)(27,31);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(18,25)(19,28)(26,31)(29,30);(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,26,25,31)(19,29,28,30)(23,32,27,24)Polycyclic presentation:
2^9|b^2=h,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283675, #72283690, #72284281, … |
| maximal normal subgroup of | #72283675, #72283690, #72284281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897461);