Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16)(7,17,19,32)(8,22)(9,14,10,15)(12,21,28,20)(13,25)(18,29)(23,36)(24,31)(26,30)(27,35)(33,34);(5,7)(6,12,16,28)(8,23,25,35)(9,20)(10,21)(11,19)(13,29)(14,32,15,17)(18,24,34,26)(22,33)(27,31)(30,36);(1,3,4,2)(5,8,9,24,11,25,10,26)(6,13,14,30,16,22,15,31)(7,18,20,23,19,34,21,35)(12,27,32,29,28,36,17,33);(6,14,16,15)(7,19)(12,17,28,32)(13,30,22,31)(18,34)(20,21)(23,35)(27,33,36,29)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=gh,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280083, #72280622 |
| maximal normal subgroup of | #72280083, #72280622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282771);