Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,17)(15,19,20,18)(21,22)(23,28)(24,32)(25,30)(26,27)(29,35)(31,34)(33,36);(5,7,9,12)(6,8,10,11)(14,17)(15,20)(22,27)(25,28)(29,32)(34,36);(6,10)(8,11)(13,15,16,20)(14,18,17,19)(21,23,26,30)(22,28,27,25)(24,33,35,31)(29,34,32,36);(1,2,3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,35)(22,29,27,32)(23,31,30,33)(25,34,28,36);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,25,26,28)(22,30,27,23)(24,34,35,36)(29,33,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1554247);