Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #95 |
| commutator, prime radical | C23 | C82 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1273 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16)(18,22)(19,25)(24,29)(26,31);(2,6)(3,9)(8,13)(10,15)(17,18,19,23,20,24,26,30,21,22,25,28,27,29,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ch,e^a=ei,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210253, C2×#92910, #72210675 |
| maximal normal subgroup of | #72210253, C2×#92910, #72210675 |
| maximal quotient of | #72210244, C2×#92910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106);