Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,11,16,9,13)(7,17,21,12,18,27,20,28)(8,22,23,30,25,29,24,14)(19,26,34,35,32,36,33,31);(5,7,11,18)(6,12,16,28)(8,19,25,32)(9,21,10,20)(13,27,15,17)(14,26,30,36)(22,31,29,35)(23,33,24,34);(1,2,4,3)(6,13)(7,18)(8,23)(9,10)(12,17)(15,16)(19,33)(22,29)(24,25)(27,28)(31,35)(32,34);(1,3,4,2)(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,26)(13,29)(15,22)(16,30)(17,31)(18,32)(20,33)(21,34)(27,35)(28,36)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=ef,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370590);