Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1154 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)(7,18,21,32)(8,14)(9,26)(10,27)(11,28)(12,17)(13,20,31,22)(15,25)(16,24)(19,33,23,36)(29,35,30,34);(2,4)(5,7,10,22,11,23,24,34,12,21,9,20,8,19,25,35)(6,13,16,32,14,30,27,36,17,31,15,18,28,29,26,33);(7,19,21,23)(9,10)(13,29,31,30)(15,16)(18,33,32,36)(20,35,22,34)(24,25)(26,27);(5,8,12,11)(6,14,17,28)(7,19,21,23)(9,24,10,25)(13,30,31,29)(15,26,16,27)(18,33,32,36)(20,34,22,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#493077, #72216636, #72216650, … |
| maximal normal subgroup of | C2×#493077, #72216636, #72216650, … |
| maximal quotient of | #72216458, C4×#12647, #72216468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400273);