Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| 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 | #59041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(18,23)(20,21)(24,30)(25,29)(26,27)(31,32);(10,13)(11,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,30)(27,32)(28,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,24,25,31)(20,26,21,27)(23,30,29,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(19,26)(21,22)(23,29)(27,28)(30,32);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72316766, #72316919, #72316953, … |
| maximal normal subgroup of | #72316766, #72316919, #72316953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7067967);