Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196134 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15)(8,19,21,12)(9,13)(10,24)(11,17)(14,23)(16,31,33,26)(18,27)(20,28,35,36)(22,29,25,30)(32,34);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,20)(10,18,23,34)(12,28)(14,27,24,32)(16,25)(19,36)(21,35)(22,33)(26,30)(29,31);(5,8)(6,12)(7,16)(9,21)(10,25)(11,26)(13,19)(14,30)(15,31)(17,33)(18,35)(20,34)(22,23)(24,29)(27,36)(28,32);(1,3,4,2)(6,13)(11,15)(12,19)(14,24)(26,31)(27,32)(28,36)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272167);