Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3226 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)(13,14,17,21,18,22,15,19)(16,20,24,27,25,28,23,26);(6,10)(8,11)(14,19)(15,17)(20,26)(21,22)(23,24)(27,28);(2,4)(5,6)(7,8)(9,10)(11,12)(13,15,18,17)(14,19,22,21)(16,23,25,24)(20,26,28,27);(5,7,9,12)(6,11,10,8)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28);(5,8,9,11)(6,7,10,12)(13,15,18,17)(14,21,22,19)(16,24,25,23)(20,26,28,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524822);