Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1865 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)(5,6,9,11)(7,12,8,10)(22,26)(23,29)(28,34)(31,32);(5,7)(6,10)(8,9)(11,12)(14,17)(18,20)(21,22,25,26)(23,27,29,30)(24,28,33,34)(31,35,32,36);(7,8)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,35)(33,34);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(21,23)(22,27)(24,32)(25,29)(26,30)(28,36)(31,33)(34,35);(5,6)(7,10)(8,12)(9,11)(13,15,16,19)(14,18,17,20)(21,24,25,33)(22,28,26,34)(23,31,29,32)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2671554);