Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,17,16)(15,18,19,20)(21,22,26,25)(23,28,24,27);(1,2,3,4)(5,7)(6,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(22,25)(27,28);(1,2)(3,4)(7,8)(10,11)(14,16)(18,20)(21,23,26,24)(22,27,25,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,23)(22,28,25,27);(5,8,9,7)(6,11,12,10)(13,16,17,14)(15,18,19,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=fgi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3936653);