Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,17,12,18,11,16)(7,19,22,13,24,26,23,34)(8,25,27,30,29,21,28,14)(9,15,32,35,33,20,31,36);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,26,18,34,16,19)(8,20,28,35,29,15,27,36)(9,30,32,21,33,14,31,25);(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,31)(15,24)(16,21)(17,25)(18,30)(19,33)(22,36)(23,35)(32,34);(2,4)(5,9,10,32,12,33,11,31)(6,15,17,35,18,20,16,36)(7,21,22,14,24,25,23,30)(8,26,27,34,29,19,28,13)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,f^c=fi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314246 |
| maximal normal subgroup of | #72314246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263697);