Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,19)(16,20)(17,18)(22,25)(23,27);(2,4)(5,7,9,8)(6,10,12,11)(13,14,17,18)(15,19,20,16)(23,27)(26,28);(1,2,3,4)(7,8)(10,11)(14,18)(15,20)(21,22,24,25)(23,26,27,28);(5,8,9,7)(6,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,23,24,27)(22,26,25,28);(5,6)(7,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=hi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 844908);