Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(14,18)(15,20)(21,22,26,24)(23,28,25,27);(6,10)(8,11)(13,14,17,18)(15,19,20,16)(21,23)(22,27)(24,28)(25,26);(1,2)(3,4)(21,22,26,24)(23,28,25,27);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,28,25,27);(5,8,9,11)(6,12,10,7)(13,16,17,19)(14,20,18,15)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 701406);