Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1445 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)(5,6,11,17)(7,18)(8,25,28,21)(9,26)(10,15,19,16)(12,23)(13,31,34,27)(14,32)(20,30)(22,33)(24,36)(29,35);(1,2,4,3)(5,7,19,22,11,24,10,23)(6,12,16,18,17,33,15,36)(8,26,31,35,28,20,27,14)(9,21,29,13,30,25,32,34);(5,8,11,28)(6,13,17,34)(7,20,24,26)(9,12,30,33)(10,31,19,27)(14,22,35,23)(15,25,16,21)(18,29,36,32);(1,3,4,2)(5,9)(6,14)(7,21)(8,12)(10,32)(11,30)(13,22)(15,20)(16,26)(17,35)(18,31)(19,29)(23,34)(24,25)(27,36)(28,33)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=efgi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279936);