Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| 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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 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)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,31,26,32);(3,8)(5,7)(10,13)(14,15)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(2,6)(5,7);(9,11,12,16)(10,14,13,15)(19,28)(23,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72256909, #72256940, #72256993 |
| maximal normal subgroup of | #72256909, #72256940, #72256993 |
| maximal quotient of | #72264579 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6709899);