Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2318 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)(7,12)(8,9)(10,11)(14,18)(15,20);(1,3,4,2)(6,11)(7,8)(22,25)(26,28);(5,7,10,8)(6,12,11,9)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(5,8,10,7)(6,9,11,12)(13,15,17,20)(14,19,18,16);(5,9)(6,7)(8,11)(10,12)(13,16,17,19)(14,20,18,15)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 429636);