Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,15,12,9,10,16)(7,17,20,28,21,30,19,13)(8,22,25,24,27,32,26,14)(18,29,23,31,35,36,34,33);(1,2,3,4)(5,7,10,19,12,21,11,20)(6,13,15,17,9,28,16,30)(8,23,25,35,27,34,26,18)(14,31,32,29,24,33,22,36);(5,8,12,27)(6,14,9,24)(7,18,21,35)(10,25,11,26)(13,29,28,36)(15,32,16,22)(17,33,30,31)(19,23,20,34);(2,4)(5,9)(6,12)(7,17)(8,24)(10,15)(11,16)(13,19)(14,27)(18,33)(20,28)(21,30)(22,26)(23,29)(25,32)(31,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326239);