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×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1659 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(2,8)(7,15)(9,13)(10,16)(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(19,27)(20,29)(23,26)(28,31);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)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,f^a=fh,d^b=di,f^b=fh,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72286868, #72286982, #72287143, … |
| maximal normal subgroup of | #72286868, #72286982, #72287143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252313);