Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,20,24)(15,17,23,22)(16,28,27,21)(18,26,19,25);(1,2,3,4)(7,8)(10,12)(13,15,19,16,20,23,18,27)(14,21,26,22,24,28,25,17);(5,7,9,8)(6,10,11,12)(13,16,19,23,20,27,18,15)(14,22,26,28,24,17,25,21);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,24)(16,25)(18,28)(19,21)(20,22)(26,27);(6,11)(10,12)(14,24)(17,22)(21,28)(25,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390663);