Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,11)(7,10,12,9)(13,14,17,20)(15,19,16,18)(21,22,25,26)(23,27,28,24);(5,7)(6,9)(8,12)(10,11)(15,16)(18,19)(22,26)(24,27);(7,12)(9,10)(13,15,17,16)(14,18,20,19)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,19,20,18)(21,23)(22,24)(25,28)(26,27);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fg,c^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374221);