Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2782 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,8,11,14,16,22)(7,15,19,29,20,17,26,12)(9,23,24,21,25,13,27,30)(18,31,34,28,35,32,36,33);(1,2)(3,4)(5,7,16,19,11,20,10,26)(6,12,22,15,14,29,8,17)(9,18,24,36,25,35,27,34)(13,28,30,31,23,33,21,32);(2,4)(5,8,10,14,11,22,16,6)(7,17,19,12,20,15,26,29)(9,13,24,30,25,23,27,21)(18,33,34,31,35,28,36,32);(5,9)(6,13)(7,18)(8,21)(10,27)(11,25)(12,28)(14,23)(15,31)(16,24)(17,32)(19,36)(20,35)(22,30)(26,34)(29,33)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363229);