Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,21,17,22)(15,24,25,26,20,27,28,16);(2,3)(13,15,19,20)(14,16,21,26)(17,25,18,28)(22,24,23,27);(1,2,4,3)(6,9)(10,12)(13,16)(14,20)(15,21)(17,27)(18,24)(19,26)(22,25)(23,28);(1,3,4,2)(14,21)(16,26)(22,23)(24,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,21,23)(15,25,20,28)(16,24,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374610);