Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #23863 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 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)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,29)(21,24)(22,25)(26,30)(27,31)(28,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,30)(25,32)(29,31);(1,3,5,4)(2,7,6,8)(10,14)(11,15);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,26)(20,21)(23,30)(24,29)(27,28)(31,32);(2,6)(7,8)(17,20,22,21)(18,24,25,29)(19,27,28,26)(23,30,32,31)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72308943, #72308985 |
| maximal normal subgroup of | #72308943, #72308985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433012);