Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 |
| prime radical | C2×#3522 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3164 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)(13,14,18,19,17,22,15,20)(16,24,25,28,26,21,23,27);(6,10)(8,11)(14,19)(15,18)(20,22)(21,27)(23,25)(24,28);(2,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,18)(14,20,22,19)(16,23,26,25)(21,28,24,27);(5,7,9,12)(6,11,10,8)(13,16)(14,21)(15,23)(17,26)(18,25)(19,27)(20,28)(22,24);(5,8,9,11)(6,7,10,12)(13,15,17,18)(14,20,22,19)(16,25,26,23)(21,27,24,28)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524766);