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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2010 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)(8,10)(9,12)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36);(5,6,8,10)(7,9,12,11)(14,17)(18,20)(22,26)(23,30)(24,33)(34,35);(7,12)(9,11)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,18,17,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,e^2=h,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, 1415498);