Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,29,31,24)(25,28,33,34)(30,35,36,32);(1,2,3,4)(6,9)(10,12)(15,19)(17,20)(22,27)(24,29)(25,33)(30,36);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,31)(22,24,27,29)(25,30,33,36)(28,32,34,35);(13,15,16,19)(14,17,18,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,35)(34,36);(5,7,8,11)(6,10,9,12)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fg,e^2=h,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464107);