Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,18)(15,19)(23,27)(24,26);(1,3,4,2)(6,9)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22,25,28)(23,24,27,26);(7,12)(10,11)(14,18)(15,19)(21,23,25,27)(22,26,28,24);(5,7)(6,10)(8,12)(9,11)(13,15,17,19)(14,16,18,20)(21,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,18)(17,20)(21,24,25,26)(22,27,28,23)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 469177);