Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3158 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)(7,12)(8,10)(9,11)(14,18)(15,20)(22,26)(23,24);(5,7,9,8)(6,10,11,12)(13,14,17,18)(15,16,20,19)(21,22,25,26)(23,28,24,27);(13,15)(14,16)(17,20)(18,19)(21,23,25,24)(22,27,26,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,26);(5,8,9,7)(6,10,11,12)(13,16,17,19)(14,20,18,15)(21,24,25,23)(22,27,26,28)Polycyclic presentation:
2^9|b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769686);