Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #18158 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,22)(17,28)(20,34)(21,33)(23,31)(24,38)(26,30)(27,32)(29,35)(36,40)(37,39);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,33)(18,30)(19,25)(22,28)(23,36)(24,37)(27,35)(31,39)(32,40)(34,38);(9,12,14,23)(10,16,18,31)(11,20,21,36)(13,27,28,24)(15,29,30,39)(17,32,25,38)(19,34,33,40)(22,37,26,35);(1,3,5,4)(2,7,8,6)(11,21)(12,23)(15,30)(16,31)(19,33)(22,26)(24,27)(32,38)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72281069 |
| maximal normal subgroup of | #72281069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393132);