Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,17,9,16)(7,18,22,31,23,30,21,12)(8,14,26,33,27,32,25,13)(19,28,24,36,20,34,35,29);(1,2)(3,4)(5,7,9,21,11,23,10,22)(6,12,15,18,17,31,16,30)(8,24,25,19,27,35,26,20)(13,29,14,28,33,36,32,34);(2,4)(5,8)(6,13)(7,19)(9,26)(10,25)(11,27)(12,28)(14,16)(15,32)(17,33)(18,29)(20,23)(21,24)(22,35)(30,36)(31,34);(5,8)(6,14,17,32)(7,20)(9,26)(10,25)(11,27)(12,29,31,36)(13,16,33,15)(18,34,30,28)(19,23)(21,35)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 336065);