Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2365 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)(7,17)(8,22,24,13)(9,16)(10,14)(11,15)(12,20)(18,32,35,31)(19,28)(21,27)(23,29,25,30)(26,34,36,33);(1,2)(3,4)(5,7,10,19,11,21,9,20)(6,12,16,27,15,28,14,17)(8,18,25,33,24,35,23,34)(13,26,30,32,22,36,29,31);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(2,4)(7,19)(9,10)(12,27)(14,16)(17,28)(18,33)(20,21)(23,25)(26,32)(29,30)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280827, #72280919, #72281044, … |
| maximal normal subgroup of | #72280827, #72280919, #72281044, … |
| maximal quotient of | #72313254, #99×#102 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263300);