Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4022 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)(7,12,18,27)(8,20,23,28)(9,14)(10,24)(11,15)(13,22,29,21)(16,30)(17,26,19,34)(25,31)(32,35)(33,36);(2,4)(5,7,10,19,11,18,9,17)(6,12,14,26,15,27,24,34)(8,16,22,33,23,32,21,31)(13,25,28,35,29,36,20,30);(5,8)(6,13)(7,16)(9,21)(10,22)(11,23)(12,25)(14,28)(15,29)(17,31)(18,32)(19,33)(20,24)(26,35)(27,36)(30,34);(5,9,11,10)(6,14,15,24)(7,17,18,19)(8,21,23,22)(12,26,27,34)(13,28,29,20)(16,31,32,33)(25,35,36,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239603, #72239611, #72239647, … |
| maximal normal subgroup of | #72239603, #72239611, #72239647, … |
| maximal quotient of | #72239692, C2×#353465, #72316882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260661);