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 | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,9)(8,10)(11,12)(13,14,17,15)(16,18,20,19)(22,26)(27,32)(28,33)(34,36);(1,2,3,4)(5,6,8,10)(7,11,12,9)(13,15,17,14)(16,18,20,19)(23,29)(27,32)(30,35)(34,36);(5,6,8,10)(7,9,12,11)(13,16)(14,18)(15,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,6)(7,9)(8,10)(11,12)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7,8,12)(6,9,10,11)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=fg,e^2=g,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, 464015);