Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 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)(5,6,11,10)(7,8,9,12)(13,14,17,19)(15,21,22,28)(16,24,25,20)(18,26,27,23);(5,7,11,9)(6,12,10,8)(13,15,16,23,17,22,25,26)(14,18,20,21,19,27,24,28);(1,2,3,4)(5,8,11,12)(6,7,10,9)(14,19)(15,22)(20,24)(23,26);(5,9,11,7)(6,12,10,8)(15,22)(18,27)(21,28)(23,26);(5,10,11,6)(7,8,9,12)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477777);