Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3638 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)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,21)(22,24)(26,30)(27,28);(10,13)(14,16)(18,20)(19,24)(25,26,29,30)(27,32,28,31);(11,15)(14,16)(18,20)(22,23)(25,27,29,28)(26,31,30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,23)(21,24)(25,28,29,27)(26,32,30,31);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,22,24,23)(25,26)(27,32)(28,31)(29,30)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72260624, #72260651, #72260787, … |
| maximal normal subgroup of | #72260624, #72260651, #72260787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028752);