Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2170 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)(5,6)(7,10)(8,12)(9,11)(13,14,19,18)(15,20,25,28)(16,21,17,22)(23,26,24,27);(5,7,10,6)(8,12,9,11)(13,15,19,25)(14,20,18,28)(16,26,17,27)(21,24,22,23);(2,4)(13,16,14,21,19,17,18,22)(15,23,28,27,25,24,20,26);(5,8,10,9)(6,11,7,12)(16,17)(21,22)(23,24)(26,27);(5,9,10,8)(6,11,7,12)(13,17,18,21,19,16,14,22)(15,24,20,27,25,23,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,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,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718766);