Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | #18520 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18,23,26,24,27,22,32)(19,28,29,25,21,20,30,31);(2,7)(8,15)(9,13)(10,16)(17,19,23,29,24,21,22,30)(18,25,26,20,27,31,32,28);(1,3,5,12)(2,8,7,15)(4,11,6,14)(9,16,13,10)(17,20,24,28)(18,19,27,21)(22,25,23,31)(26,29,32,30);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,16,15,10)(17,21,24,19)(18,20,27,28)(22,29,23,30)(25,26,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dehi,e^b=eh,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287475 |
| maximal normal subgroup of | #72287475 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 135445);