Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2676 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)(6,7)(8,12)(13,14)(15,18)(16,20)(17,19)(23,27)(25,28);(5,6,9,7)(8,11,12,10)(13,15,17,20)(14,18,19,16)(21,22,24,26)(23,25,27,28);(1,3,4,2)(13,15)(14,16)(17,20)(18,19)(21,23)(22,25)(24,27)(26,28);(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514078);