Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,21,14,20,15,32)(11,22,25,12,24,17,18,27)(13,19,31,37,29,36,30,40)(23,35,38,26,28,33,34,39);(1,3)(2,5)(4,8)(6,7)(9,11,16,25,14,24,15,18)(10,17,21,27,20,22,32,12)(13,28,31,34,29,23,30,38)(19,35,37,26,36,33,40,39);(9,12,16,17,14,27,15,22)(10,18,21,11,20,25,32,24)(13,26,31,33,29,39,30,35)(19,34,37,23,36,38,40,28);(3,8)(5,7)(9,13)(10,19)(11,23)(12,26)(14,29)(15,30)(16,31)(17,33)(18,34)(20,36)(21,37)(22,35)(24,28)(25,38)(27,39)(32,40);(2,6)(5,7)(10,20)(11,24)(13,29)(18,25)(21,32)(26,39)(30,31)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314467, #72315580, #72315625 |
| maximal normal subgroup of | #72314467, #72315580, #72315625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301877);