Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,18,16)(15,22,24,28)(17,20,26,21)(19,27,25,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(2,4)(6,9)(10,11)(13,15,17,23,18,24,26,27)(14,19,21,22,16,25,20,28);(5,7,8,12)(6,10,9,11)(13,16,18,14)(15,19,24,25)(17,20,26,21)(22,27,28,23);(2,4)(13,17,18,26)(14,20,16,21)(15,23,24,27)(19,28,25,22)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592077);