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)(6,11)(7,15)(9,21)(10,20)(12,26)(13,29)(16,23)(17,34)(18,33)(19,35)(22,31)(24,28)(27,30)(32,36);(5,6)(7,12)(8,11)(9,22)(10,14)(13,19)(15,26)(16,32)(17,27)(18,24)(20,25)(21,29)(23,36)(28,33)(30,34)(31,35);(5,7)(6,12)(8,18)(9,23)(10,16)(11,24)(13,30)(14,32)(15,21)(17,20)(19,34)(22,36)(25,27)(26,29)(28,31)(33,35);(6,11)(7,16,18,17)(9,19)(12,27,24,32)(14,25)(15,23,33,34)(21,35)(26,30,28,36)Polycyclic presentation:
2^9|a^2=g,d^2=ei,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, #72250067, #72250096, … |
| maximal normal subgroup of | #72249908, #72250067, #72250096, … |
| maximal quotient of | #72265511, #72265696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401251);