Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #23939 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,26,29,32)(21,25,23,30)(24,31,27,28);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,28)(27,29)(31,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,28,22,29,21,31)(18,24,25,26,20,27,30,32);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20,22,18)(19,27,29,24)(21,25,23,30)(26,28,32,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257418, #72257520, #72264068 |
| maximal normal subgroup of | #72257418, #72257520, #72264068 |
| maximal quotient of | #72265028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6937539);