Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483023 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17675 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | #60704 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,11,5,7,19,21,6,8,17,23,20,10,16,27)(3,12,14,29,15,22,31,28,13,26,18,32,30,9,24,25);(2,7)(4,16)(5,20)(8,10)(9,12)(11,23)(14,24)(15,30)(17,19)(18,31)(22,26)(25,28);(2,8)(4,17)(7,10)(9,22)(12,26)(14,18)(16,19)(24,31);(1,3)(2,9,8,22)(4,18)(5,15)(6,13)(7,12,10,26)(11,28,23,25)(14,17)(16,31)(19,24)(20,30)(21,32,27,29)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,f^c=fi,e^d=ei,f^d=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72292716, #72292717, C2×#483023 |
| maximal normal subgroup of | #72292716, #72292717, C2×#483023 |
| maximal quotient of | C2×#483023, #72292819 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390219);