Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 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)(13,14,18,23)(15,24,19,20)(16,26,27,28)(17,22,21,25);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,19)(14,20,23,24)(16,22,27,25)(17,28,21,26);(6,9)(7,11)(13,16,19,22,18,27,15,25)(14,21,24,26,23,17,20,28);(5,7)(6,10)(8,11)(9,12)(13,17,18,21)(14,22,23,25)(15,26,19,28)(16,24,27,20);(6,9)(7,11)(16,27)(17,21)(22,25)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263894, #72263923, #72264011 |
| maximal normal subgroup of | #72263894, #72263923, #72264011 |
| maximal quotient of | #72264586, #72264858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513685);