Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37658 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1945 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,31)(26,29)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,31,32,29);(10,14)(11,15)(17,20,23,31,22,30,21,29)(18,24,25,28,19,26,27,32);(2,6)(5,7)(18,19)(24,26)(25,27)(28,32);(3,8)(5,7)(11,15)(13,16)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,28,26,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264267, #72264310, #72264334 |
| maximal normal subgroup of | #72264267, #72264310, #72264334 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669129);