Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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×#2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,17)(18,19)(22,25)(26,28);(1,3,4,2)(6,9)(10,11)(13,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28);(6,9)(10,11)(14,19)(15,16)(23,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,19,20)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,19)(18,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 452322);