Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17)(7,13)(8,21,25,34)(9,26)(11,27,12,16)(14,28,33,24)(15,19)(18,35)(20,30)(22,32)(23,29)(31,36);(5,7,12,9,10,19,11,20)(6,13,16,26,17,15,27,30)(8,22,24,18,25,23,28,36)(14,31,34,32,33,35,21,29);(5,8)(6,14)(7,18)(9,23)(10,25)(11,28)(12,24)(13,29)(15,32)(16,34)(17,33)(19,36)(20,22)(21,27)(26,31)(30,35);(2,4)(5,9,12,19,10,20,11,7)(6,15,16,30,17,13,27,26)(8,23,24,36,25,22,28,18)(14,32,34,35,33,29,21,31)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344208);