Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 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)(7,15)(8,11)(9,13,25,20)(10,21)(12,17)(14,23)(16,32,34,28)(18,33)(19,31)(22,36,35,27)(24,30,26,29);(1,3,4,2)(5,7,10,17,11,18,23,31)(6,12,14,15,8,19,21,33)(9,22,24,34,25,35,26,16)(13,28,29,36,20,32,30,27);(2,3)(5,8,11,6)(7,12,18,19)(9,20,25,13)(10,21,23,14)(15,31,33,17)(16,27,34,36)(22,28,35,32)(24,30,26,29);(2,3)(5,9,23,24,11,25,10,26)(6,13,21,29,8,20,14,30)(7,16,31,22,18,34,17,35)(12,27,33,28,19,36,15,32)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=i,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343473);