Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,29,24,26,14,12,16,25,28,11,8,9,23)(7,17,21,13,30,32,20,18,19,35,36,31,22,34,27,33);(5,7,11,30,12,19,29,22)(6,13,24,33,16,31,8,18)(9,21,25,20,26,36,10,27)(14,34,28,35,23,32,15,17);(2,4)(5,8)(6,11)(7,18)(9,23)(10,28)(12,24)(13,30)(14,26)(15,25)(16,29)(17,20)(19,33)(21,32)(22,31)(27,35)(34,36);(7,19)(13,31)(17,35)(18,33)(20,27)(21,36)(22,30)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249997, #72250005, #72250042, … |
| maximal normal subgroup of | #72249997, #72250005, #72250042, … |
| maximal quotient of | #72264974 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401739);