Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,14)(7,17,21,28,19,32,20,12)(8,22,24,13,26,30,25,31)(18,34,23,33,35,27,36,29);(1,2)(3,4)(5,7)(6,12,16,28)(8,23,26,36)(9,20)(10,21)(11,19)(13,29)(14,32,15,17)(18,24,35,25)(22,27)(30,34)(31,33);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,22,16,31,14,30)(7,18,20,23,19,35,21,36)(12,27,17,33,28,34,32,29);(6,14,16,15)(7,19)(12,17,28,32)(13,30,31,22)(18,35)(20,21)(23,36)(27,33,34,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280664, #72280718 |
| maximal normal subgroup of | #72280664, #72280718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273466);