Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3088 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)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,13)(14,15)(17,18,21,22)(19,23,24,20)(26,29)(30,32);(2,6)(5,7)(10,13)(14,16)(18,22)(20,23)(25,27,28,31)(26,30,29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=g,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326958, #72326978, #72326992, … |
| maximal normal subgroup of | #72326958, #72326978, #72326992, … |
| maximal quotient of | #72328394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293611);