Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19,23,21,24,27,26)(20,29,28,25,31,22,30,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,22)(19,27)(20,30)(23,25)(24,29)(26,32)(28,31);(3,9)(6,12)(10,15)(13,16)(17,19,21,27)(18,23,24,26)(20,28,31,30)(22,32,29,25);(18,24)(22,29)(23,26)(25,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=g,f^2=h,g^2=i,b^a=be,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216720, #72216914, #72216933, … |
| maximal normal subgroup of | #72216720, #72216914, #72216933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388511);