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)(7,16)(8,11)(9,22,26,31)(10,15)(12,18)(13,27,21,25)(14,24)(17,28,34,36)(19,29)(20,32)(23,33,35,30);(1,3,4,2)(5,7,10,18,11,19,24,32)(6,12,14,16,8,20,15,29)(9,23,25,34,26,35,27,17)(13,30,22,36,21,33,31,28);(2,3)(5,8,11,6)(7,12,19,20)(9,21,26,13)(10,15,24,14)(16,32,29,18)(17,28,34,36)(22,25,31,27)(23,30,35,33);(2,3)(5,9,24,25,11,26,10,27)(6,13,15,22,8,21,14,31)(7,17,32,23,19,34,18,35)(12,28,29,30,20,36,16,33)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=i,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343471);