Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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×#3282 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,22,16,19)(18,21,25,28)(23,27,24,26);(6,9)(10,12)(13,15,17,16)(14,19,20,22)(18,23,25,24)(21,26,28,27);(2,4)(5,7,8,11)(6,10,9,12)(15,23)(16,24)(18,25)(19,26)(21,28)(22,27);(13,16,17,15)(14,19,20,22)(18,24,25,23)(21,26,28,27);(15,23)(16,24)(18,25)(19,26)(21,28)(22,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482018);