Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 |
| prime radical | #37618 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 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)(7,12)(8,10)(9,11)(13,14,18,22)(15,23,26,20)(16,17,21,19)(24,28,27,25);(2,4)(7,8)(10,12)(13,15)(14,20)(16,27)(17,28)(18,26)(19,25)(21,24)(22,23);(5,7,9,8)(6,10,11,12)(13,16,19,14,18,21,17,22)(15,24,25,23,26,27,28,20);(5,8,9,7)(6,10,11,12)(13,17,18,19)(14,21,22,16)(15,25,26,28)(20,27,23,24);(6,11)(10,12)(14,22)(15,26)(16,21)(25,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390347);