Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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×#3284 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2125 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,20)(15,21,17,23)(16,26,28,22)(18,24,27,25);(6,9)(10,12)(13,15)(14,21)(16,18)(17,19)(20,23)(22,25)(24,26)(27,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,28)(14,22,20,26)(15,25,17,24)(18,21,27,23);(13,17,19,15)(14,23,20,21)(16,27,28,18)(22,24,26,25);(13,18)(14,24)(15,22)(16,21)(17,26)(19,27)(20,25)(23,28)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482066);