Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347305 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,6)(3,4)(7,8)(10,13)(11,15)(14,16);(1,2)(3,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,15);(11,15)(13,16);(1,3,5,4)(2,6,7,8);(2,7)(6,8)(10,14)(13,16)Polycyclic presentation:
2^9|d^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214454, #72214761, #72214850, … |
| maximal normal subgroup of | #72214454, #72214761, #72214850, … |
| maximal quotient of | C2×#6347316, #72279229, #72283127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254512);