Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574031 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,28)(21,26)(22,23)(24,29)(27,32)(30,31);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,26,25)(20,29,30,27)(24,28,32,31);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,29,21,30,19,27)(18,24,25,28,26,32,23,31);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(20,30)(24,32)(27,29)(28,31);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,19,21,22)(18,25,26,23)(20,29,30,27)(24,31,32,28)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502703);