Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2189 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)(7,8)(9,10)(11,12)(13,14,17,20)(15,18,19,16)(22,26)(23,29)(28,34)(31,32);(6,10)(7,12)(13,15)(14,16)(17,19)(18,20)(21,22,25,26)(23,30,29,27)(24,28,33,34)(31,36,32,35);(6,10)(8,11)(13,15)(14,18)(16,20)(17,19)(21,23,25,29)(22,27,26,30)(24,32,33,31)(28,36,34,35);(1,2,3,4)(5,7,9,12)(6,11,10,8)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,36)(33,34);(5,8,9,11)(6,12,10,7)(13,16,17,18)(14,19,20,15)(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=h,c^2=h,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575309);