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 | |
2165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(15,19)(18,20)(21,22,26,27)(23,25,30,28)(24,29,34,36)(31,35,32,33);(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,16)(15,18,19,20)(22,27)(23,30)(29,36)(31,32);(1,2,3,4)(6,9)(10,12)(21,23,26,30)(22,28,27,25)(24,32,34,31)(29,35,36,33);(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,34)(22,29,27,36)(23,31,30,32)(25,35,28,33);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,25,26,28)(22,30,27,23)(24,33,34,35)(29,31,36,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgh,d^2=g,e^2=gi,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, 2698845);