Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1507 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)(7,19,22,13,24,32,23,33)(8,25,26,14,27,9,15,29)(20,21,36,35,31,30,34,28);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,32,18,33,16,19)(8,20,15,34,27,31,26,36)(9,28,29,21,25,35,14,30);(5,8,10,26,12,27,11,15)(6,14,17,9,18,29,16,25)(7,20,22,36,24,31,23,34)(13,30,32,28,33,21,19,35);(2,4)(5,9,10,29,12,25,11,14)(6,15,17,8,18,26,16,27)(7,21,22,35,24,30,23,28)(13,31,32,34,33,20,19,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,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, 263878);