Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,19)(15,16,20,18)(21,22,26,27)(23,25,24,28);(6,10)(7,12)(13,15,17,20)(14,18,19,16)(21,23,26,24)(22,25,27,28);(7,12)(8,11)(14,19)(16,18)(22,27)(23,24);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,25,27,28);(5,8)(6,7)(9,11)(10,12)(13,16,17,18)(14,15,19,20)(21,25)(22,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5189059);