Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(13,14,18,22)(15,20,16,25)(17,21,26,27)(19,23,24,28);(5,6)(7,12)(8,11)(9,10)(13,15,18,16)(14,20,22,25)(17,19,26,24)(21,23,27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,26,18,15,24,17)(14,20,23,21,22,25,28,27);(5,8,9,7)(6,11,10,12);(13,17,19,16,18,26,24,15)(14,21,23,25,22,27,28,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717854);