Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1316 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)(5,6,9,17,10,16,11,15)(7,18,20,12,21,13,27,29)(8,22,24,14,25,33,26,32)(19,34,35,31,23,28,36,30);(5,7,9,20,10,21,11,27)(6,12,15,18,16,29,17,13)(8,19,24,35,25,23,26,36)(14,28,22,31,32,34,33,30);(1,2,3,4)(5,7,10,21)(6,13,16,18)(8,23,25,19)(9,20,11,27)(12,17,29,15)(14,31,32,30)(22,34,33,28)(24,36,26,35);(5,8,9,24,10,25,11,26)(6,14,15,22,16,32,17,33)(7,19,20,35,21,23,27,36)(12,28,18,31,29,34,13,30)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267971);