Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | #61124 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,14,19,15,17,13,28)(11,20,22,16,23,32,29,30)(12,18,27,35,25,33,26,40)(21,36,37,31,24,38,39,34);(1,2)(3,6)(4,7)(5,8);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,24)(13,29)(14,22)(15,23)(17,30)(18,34)(19,32)(20,28)(21,25)(26,37)(27,39)(31,33)(35,36)(38,40);(1,3,5,4)(2,7,8,6)(10,17)(12,25)(16,30)(19,28)(20,32)(21,24)(26,27)(37,39);(1,4,5,3)(2,7,8,6)(9,12,14,27,15,25,13,26)(10,18,19,35,17,33,28,40)(11,21,22,37,23,24,29,39)(16,31,32,38,30,34,20,36)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72314315, #72315607, #72315625 |
| maximal normal subgroup of | #72314315, #72315607, #72315625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307683);