Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,31)(30,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(19,26)(22,30)(25,31)(27,32);(9,11)(10,13)(12,16)(14,15)(17,19,20,27,21,26,29,32)(18,22,24,25,23,30,28,31);(1,2,5,4)(3,7,8,6)(11,16)(13,15);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,23)(19,26)(24,28)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251184, #72256852, #72256909 |
| maximal normal subgroup of | #72251184, #72256852, #72256909 |
| maximal quotient of | #72264950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6938251);