Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,8)(6,7)(9,10,14,16,13,11,12,15);(1,3,4,2,6,8,5,7)(9,11,13,10)(12,15,14,16);(12,14)(15,16);(1,4,6,5)(2,3,7,8)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=h,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214454, #72214750, #72214761, … |
| maximal normal subgroup of | #72214454, #72214750, #72214761, … |
| maximal quotient of | C2×#200521, #72282907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107717);