Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1060 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)(3,4)(5,6,8,14,9,15,22,31,10,13,21,29,23,12,27,28)(7,16,18,30,19,24,33,35,17,11,20,25,32,26,34,36);(5,7)(6,11)(8,20)(9,19)(10,17)(12,24)(13,16)(14,30)(15,26)(18,21)(22,34)(23,32)(25,29)(27,33)(28,36)(31,35);(2,4)(7,17)(11,16)(18,20)(19,32)(24,26)(25,30)(33,34)(35,36);(6,12,13,15)(8,21)(11,24,16,26)(14,31,29,28)(18,20)(22,27)(25,36,30,35)(33,34)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=ci,g^b=gi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239289, #72250086, #72250090, … |
| maximal normal subgroup of | #72239289, #72250086, #72250090, … |
| maximal quotient of | #72239474, #72239479 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401145);