Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,13,16)(4,8)(5,9)(6,10)(11,14,12,15)(17,18)(19,22,26,30)(20,27,28,23)(21,24)(25,29)(31,32);(1,3)(2,7,10,16)(4,11)(5,12)(6,13)(8,14,9,15)(17,19)(18,22,24,30)(20,25)(21,26)(23,29,27,32)(28,31);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20,21,28)(18,23,24,27)(19,25,26,31)(22,29,30,32);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251118, #72251255, #72263643, … |
| maximal normal subgroup of | #72251118, #72251255, #72263643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108859);