Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234453 |
| 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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1566 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,12,13,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31);(2,6)(3,4)(9,11,14,13)(10,12,15,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32);(1,2)(3,8)(4,7)(5,6)(9,12,13,15,14,16,11,10);(1,3,5,4)(2,7,6,8)(9,11,14,13)(10,12,15,16)(19,26)(22,25)(27,32)(30,31);(1,4,5,3)(2,7,6,8)(9,11,14,13)(10,12,15,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7199993);