Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481819 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,27)(20,31)(21,34,36,25)(22,26)(23,30,38,35)(24,33)(28,42)(29,40)(32,39,48,46)(37,47,45,44)(41,43);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23,26,38)(20,28)(21,35)(22,29)(24,37,31,45)(25,40,34,27)(30,36)(32,47)(33,43)(39,42,46,41)(44,48);(2,7)(6,14)(8,15)(9,16)(17,20,22,33)(18,24,26,31)(19,28,29,43)(21,32,36,48)(23,37,38,45)(25,39,34,46)(27,41,40,42)(30,44,35,47);(19,29)(23,38)(27,40)(28,43)(30,35)(37,45)(41,42)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=hi,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284601, #72284618, #72284683, … |
| maximal normal subgroup of | #72284601, #72284618, #72284683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389054);