Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,7,13,16,11,14,12,15)(17,18)(19,24)(20,25)(21,29,30,23)(22,26)(27,32,28,31);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(18,23)(24,31)(25,32)(26,29);(3,11)(7,14)(12,13)(15,16)(17,19,22,20)(18,24,26,25)(21,27,30,28)(23,31,29,32);(1,4,6,5)(2,8,10,9)(3,12,11,13)(7,15,14,16)(17,20,22,19)(18,25,26,24)(21,28,30,27)(23,32,29,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fg,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284994, #72285018, #72285243, … |
| maximal normal subgroup of | #72284994, #72285018, #72285243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387776);