Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234958 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35129 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1466 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)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(6,13)(14,15)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(1,3)(2,6)(4,11)(5,10)(7,15)(8,13)(9,12)(14,16)(20,30)(23,28)(25,32)(26,29);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,14,13,15)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286939, #72287128, #72287135 |
| maximal normal subgroup of | #72286939, #72287128, #72287135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142166);