Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,26,30)(3,15,16,29,17,24,31,19)(4,10,20,11,21,28,5,22)(6,14,25,23,9,27,18,32);(1,3,4,13,6,10,19,27)(2,9,28,29,14,8,17,21)(5,23,18,31,24,12,26,22)(7,11,20,32,25,16,15,30);(5,24)(7,25)(11,16)(12,23)(15,20)(18,26)(22,31)(30,32);(1,4,6,19)(2,10,14,3)(5,18,24,26)(7,20,25,15)(8,21,9,29)(11,23,16,12)(13,28,27,17)(22,32,31,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213691, #72213763, #72213796, … |
| maximal normal subgroup of | #72213691, #72213763, #72213796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107647);