Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2856 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)(7,12)(8,10)(9,11)(13,14,18,19)(15,20,16,21)(17,25,28,22)(23,26,24,27);(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,20,19,21)(17,26,28,27)(22,23,25,24);(2,4)(14,19)(15,21)(16,20)(22,25)(23,26)(24,27);(5,8,9,7)(6,10,11,12)(13,16,18,15)(14,21,19,20)(17,27,28,26)(22,24,25,23);(13,17,18,28)(14,22,19,25)(15,23,16,24)(20,27,21,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,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, 6483803);