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 | |
1982 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)(5,6,9,8)(7,11,12,10)(13,14,18,16)(15,19,17,20)(21,22)(23,29)(24,31)(25,33)(26,27)(28,34)(30,32)(35,36);(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,19,16,20)(22,27)(24,29)(25,34)(30,36);(5,7,9,12)(6,11,8,10)(13,16,18,14)(15,19,17,20)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,17,18,15)(14,19,16,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8,9,6)(7,10,12,11)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=gh,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, 1554246);