Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 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)(7,11)(8,12)(9,10)(14,19)(20,28)(21,23)(22,27);(6,10)(11,12)(13,14)(15,23)(16,20)(17,19)(18,27)(21,25)(22,26)(24,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,17,25)(14,20,19,28)(16,18,24,26)(21,27,23,22);(5,7,9,8)(6,11,10,12)(15,24)(16,25)(18,26)(20,21)(22,27)(23,28);(5,8,9,7)(6,12,10,11)(13,16)(14,21)(15,18)(17,24)(19,23)(20,27)(22,28)(25,26)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284950, #72285049 |
| maximal normal subgroup of | #72284950, #72285049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492709);