Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | #14762 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,8)(9,11)(10,12)(13,14,18,20)(15,19,17,16)(21,22,24,26)(23,27,28,25);(2,3)(5,7,9,12)(6,10,11,8)(13,15)(14,16)(17,18)(19,20);(13,16)(14,17)(15,20)(18,19)(21,22)(23,25)(24,26)(27,28);(2,3)(5,6)(7,10)(8,12)(9,11)(13,16)(14,15)(17,20)(18,19)(21,23,24,28)(22,25,26,27);(1,3,4,2)(5,8,9,10)(6,7,11,12)(13,17,18,15)(14,19,20,16)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=fgi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1359247);