Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2228 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)(6,9)(7,11)(13,14,17,20)(15,19,16,18)(21,22)(23,28)(24,32)(25,30)(26,27)(29,34)(31,36)(33,35);(6,9)(10,12)(13,14,17,20)(15,18,16,19)(22,27)(25,35)(28,36)(29,32);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,18,20,19)(21,23)(22,28)(24,33)(25,29)(26,31)(27,36)(30,34)(32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,34)(22,29,27,32)(23,30,31,33)(25,28,35,36);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,25,26,35)(22,30,27,33)(23,29,31,32)(24,28,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=fg,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, 1620037);