Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 |
| prime radical | C2×#3275 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(14,20)(15,25)(16,17)(24,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,27)(24,26)(25,28);(1,2,3,4)(6,9)(10,11)(13,15)(14,21)(16,19)(17,26)(18,25)(20,22)(23,27)(24,28);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,22,20,21)(15,26,25,19)(23,24,28,27);(13,17,18,16)(14,23,20,28)(15,26,25,19)(21,27,22,24)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591933);