Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232749 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #34609 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,12,15)(4,9,5,8)(7,14,16,13)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(22,31)(25,32);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(3,12)(7,16)(11,15)(13,14)(19,26)(22,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279567, #72302138, #72303170, … |
| maximal normal subgroup of | #72279567, #72302138, #72303170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140810);