Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
953 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)(7,15)(8,19)(9,23)(11,24)(13,28)(14,27)(16,21)(17,32)(18,33)(20,34)(22,35)(25,36)(26,31)(29,30);(5,6)(7,11)(8,20)(9,14)(10,12)(13,22)(15,31)(16,30)(17,26)(18,25)(19,34)(21,36)(23,27)(24,32)(28,35)(29,33);(5,7)(6,11)(8,21)(9,16)(10,17)(12,26)(13,29)(14,30)(15,19)(18,23)(20,36)(22,33)(24,28)(25,27)(31,34)(32,35);(6,12)(7,16,17,18)(8,22)(11,25,26,30)(14,27)(15,21,32,33)(19,35)(24,36,31,29)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249908, #72250062, #72250101, … |
| maximal normal subgroup of | #72249908, #72250062, #72250101, … |
| maximal quotient of | #72265511, #72265696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401250);