Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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×#2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2110 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)(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(11,12)(14,15)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(2,6)(5,7)(18,24)(22,26)(23,25)(30,31);(9,12,13,11)(10,15,16,14)(19,27)(22,26)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256896, #72256922, #72256993, … |
| maximal normal subgroup of | #72256896, #72256922, #72256993, … |
| maximal quotient of | #72308858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6755528);