Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #30529 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(18,22)(19,26)(23,25)(27,32);(1,3,4,8)(2,5,6,7)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(10,14)(15,16)(18,23)(19,26)(21,28)(22,25)(24,31)(29,30);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,19,20,26)(18,24,22,29)(21,27,28,32)(23,30,25,31);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,25)(21,28)(22,23)(24,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72285539 |
| maximal normal subgroup of | #72285539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6637946);