Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,32,26,31,24,29)(23,33,34,36,30,28,35,25);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,30)(22,28,31,33)(24,35,27,34)(25,29,36,32);(2,4)(23,30)(25,36)(28,33)(34,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,26,27)(22,29,31,32)(23,34,30,35)(25,33,36,28);(6,12)(10,13)(11,20)(18,19)(21,25,27,33,26,36,24,28)(22,30,32,35,31,23,29,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307167);