Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,23,26,31)(20,22,28,25)(27,29,32,30);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15);(11,16)(13,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,19,21,26)(18,23,24,31)(20,27,28,32)(22,29,25,30);(1,4,5,3)(2,6,7,8)(18,24)(19,26)(22,25)(27,32)Polycyclic presentation:
2^9|a^2=fi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6772727);