Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | #2018 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,16)(17,18,22,25,21,20,24,29)(19,23,28,31,26,27,30,32);(1,2,4,6,5,8,12,14)(3,7,10,13,11,15,9,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,30)(25,31)(29,32);(1,3)(2,7)(4,10)(5,11)(6,13)(8,15)(9,12)(14,16)(18,20)(19,26)(25,29)(28,30);(2,8)(6,14)(7,15)(13,16)(17,20,22,29,21,18,24,25)(19,27,28,32,26,23,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dh,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319984, C2×#211085 |
| maximal normal subgroup of | #72319984, C2×#211085 |
| maximal quotient of | C2×#211085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118281);