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×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1519 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)(5,6,9,15,11,16,10,14)(7,17,19,29,20,28,26,12)(8,21,24,13,25,31,23,30)(18,33,35,36,22,27,34,32);(1,2,4,3)(5,7)(6,12)(8,22,25,18)(9,26)(10,19)(11,20)(13,27,30,33)(14,17)(15,28)(16,29)(21,32,31,36)(23,34,24,35);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,27)(14,21)(15,31)(16,30)(17,32)(19,35)(20,22)(26,34)(28,36)(29,33);(1,3,4,2)(6,14)(9,10)(12,17)(13,21)(15,16)(19,26)(23,24)(27,32)(28,29)(30,31)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=efg,e^2=i,f^2=i,h^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, 279938);