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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,20)(15,17,16,19)(21,22,26,27)(23,24,28,25);(5,6,8,10)(7,9,12,11)(13,15,18,16)(14,19,20,17)(21,23,26,28)(22,25,27,24);(1,2,4,3)(13,15,18,16)(14,17,20,19)(21,24,26,25)(22,23,27,28);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,17,20,19)(21,25,26,24)(22,23,27,28);(1,3,4,2)(5,6)(7,11)(8,10)(9,12)(13,17)(14,16)(15,20)(18,19)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=hi,d^2=fgh,e^2=gh,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, 1400036);