Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,10,8)(7,12,9,11)(14,18)(16,19)(22,25)(26,28);(1,3,4,2)(5,7,10,9)(6,11,8,12)(13,14,17,18)(15,19,20,16);(6,8)(11,12)(13,15,17,20)(14,16,18,19)(21,22)(23,26)(24,25)(27,28);(5,8,10,6)(7,12,9,11)(13,15)(14,19)(16,18)(17,20)(21,23,24,27)(22,26,25,28);(5,9,10,7)(6,12,8,11)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423837);