Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1799 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)(5,6,9,12)(7,8,11,10)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,31,32,28)(24,33,34,29)(25,35,36,30);(5,7,9,11)(6,10,12,8)(13,15,18,17)(14,19,16,20)(22,27)(23,32)(29,33)(30,35);(5,6,9,12)(7,10,11,8)(13,16,18,14)(15,19,17,20)(21,23,26,32)(22,28,27,31)(24,30,34,35)(25,33,36,29);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,19,16,20)(21,24,26,34)(22,29,27,33)(23,30,32,35)(25,31,36,28);(5,8,9,10)(6,11,12,7)(21,25)(22,30)(23,29)(24,28)(26,36)(27,35)(31,34)(32,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375208);