Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 | #24208 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,20,25,21,23,28,22)(19,24,26,31,27,30,32,29);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(18,22)(19,26)(20,28)(23,25)(27,32)(29,31);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(18,23)(19,27)(22,25)(26,32);(3,8)(5,7)(10,11)(14,15)(17,19,20,26,21,27,28,32)(18,24,25,31,23,30,22,29);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^9|a^2=fh,c^2=h,d^2=fgi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282738, #72283778, #72284195 |
| maximal normal subgroup of | #72282738, #72283778, #72284195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104659);