Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60890 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(20,24)(21,27)(23,30)(25,33)(28,34)(29,31);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,20)(21,28)(22,24)(23,26)(25,29)(27,33)(30,32)(31,34);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21,22,29)(20,25,24,28)(23,31,30,27)(26,33,32,34);(4,10)(8,11)(9,18)(16,17)(19,20)(21,25)(22,24)(23,26)(27,34)(28,29)(30,32)(31,33);(20,24)(25,28)(26,32)(33,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482933);