Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1483 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,30)(26,28);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,22)(19,26)(21,27)(23,31)(25,28)(29,32);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,18,20,22)(19,23,25,29)(21,24,27,30)(26,32,28,31);(10,14)(13,16)(17,19)(18,23)(20,25)(21,28)(22,29)(24,31)(26,27)(30,32);(2,7)(6,8)(18,22)(19,25)(24,30)(26,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257502, #72257553, #72264215 |
| maximal normal subgroup of | #72257502, #72257553, #72264215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7431431);