Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61433 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9,7,3,5,11)(4,10,13,16,14,8,12,15)(17,18,22,19,21,23,29,25)(20,24,28,26,27,30,32,31);(1,3,7,2)(4,8,14,10)(5,9,6,11)(12,16,13,15)(18,23)(19,25)(24,30)(26,31);(1,4,5,12,7,14,6,13)(2,8,11,16,3,10,9,15)(17,19,22,23,21,25,29,18)(20,26,28,30,27,31,32,24);(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32);(20,27)(24,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,c^b=ci,f^b=fh,e^d=ehi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72283751, #72283802 |
| maximal normal subgroup of | #72283751, #72283802 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253171);