Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,17)(5,11)(6,21)(7,12)(8,14)(9,16)(10,19)(15,27)(18,24)(20,28)(22,25)(23,32)(26,30)(29,31);(1,3,8,7)(2,9,14,25)(4,15,20,19)(5,16,6,22)(10,24,27,32)(11,12,21,13)(17,26,28,31)(18,29,23,30);(1,4)(2,10)(3,15)(5,18)(6,23)(7,19)(8,20)(9,24)(11,26)(12,28)(13,17)(14,27)(16,29)(21,31)(22,30)(25,32);(1,5,8,6)(2,9,14,25)(3,16,7,22)(4,18,20,23)(10,24,27,32)(11,13,21,12)(15,29,19,30)(17,31,28,26)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72234103, C2×#511834, #72340414, … |
| maximal normal subgroup of | #72234103, C2×#511834, #72340414, … |
| maximal quotient of | #72234087, #72260961, #72261670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419030);