Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,20,23,13)(9,24)(10,26)(11,16)(12,19)(14,25)(15,27)(18,31,29,34)(21,32,28,35)(22,30,36,33);(1,2,3,4)(5,7,11,14)(6,12,16,9)(8,18)(10,24,27,19)(13,21)(15,25,26,17)(20,28)(22,35)(23,29)(30,31)(32,36)(33,34);(2,4)(5,8)(6,13)(7,18)(9,21)(10,22)(11,23)(12,28)(14,29)(15,30)(16,20)(17,31)(19,32)(24,35)(25,34)(26,33)(27,36);(5,9,11,12)(6,14,16,7)(8,21,23,28)(10,17,27,25)(13,29,20,18)(15,19,26,24)(22,31,36,34)(30,32,33,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273920);