Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,25,23,19,21,27)(20,26,30,32,28,24,29,31);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19,23,18)(20,26,28,24)(21,25,22,27)(29,31,30,32);(3,9)(6,12)(10,15)(13,16)(18,19)(24,26)(25,27)(31,32);(2,7)(6,12)(8,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,28)(25,31)(27,32);(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,f^a=fh,c^b=ci,f^b=fh,e^d=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250234);