Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10,7,3,5,9)(4,11,13,15,14,8,12,16)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(1,4,6,13,7,14,5,12)(2,8,10,16,3,11,9,15)(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(2,3)(8,11)(9,10)(15,16)(19,27)(20,29)(23,26)(28,31);(1,5,7,6)(2,9,3,10)(4,12,14,13)(8,15,11,16)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,d^b=di,f^b=fh,d^c=dhi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72321516, #72321527, #72321548, … |
| maximal normal subgroup of | #72321516, #72321527, #72321548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252335);