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 |
| prime radical | #25761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1943 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)(14,18)(19,20)(21,26)(22,28);(6,10)(11,12)(13,14,16,18)(15,22,24,28)(17,26,25,21)(19,27,20,23);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,16,24)(14,19,18,20)(17,27,25,23)(21,28,26,22);(5,8,9,7)(6,12,10,11)(15,23)(17,25)(19,22)(20,28)(21,26)(24,27);(13,15,16,24)(14,20,18,19)(17,27,25,23)(21,22,26,28)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284913, #72285147 |
| maximal normal subgroup of | #72284913, #72285147 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492069);