Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201701 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35076 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1569 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)(3,6,10,14)(4,7)(5,8)(9,13,11,15)(12,16)(17,18)(19,22,26,30)(20,27,28,23)(21,24)(25,29)(31,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,15,16,13)(9,12)(17,19)(18,22,24,30)(20,25)(21,26)(23,29,27,32)(28,31);(17,20,21,28)(18,23,24,27)(19,25,26,31)(22,29,30,32);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(18,24)(19,26)(23,27)(25,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238026, #72251255, #72263646, … |
| maximal normal subgroup of | #72238026, #72251255, #72263646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108909);