Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| 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 | #38050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 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)(14,18)(15,21)(19,23)(20,25)(22,27)(26,28);(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(6,9)(7,11)(13,15,14,19,17,23,18,21)(16,22,20,26,24,28,25,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27);(6,9)(7,11)(15,23)(16,24)(19,21)(20,25)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513696);