Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(18,22)(20,27)(23,28)(24,29)(26,31)(30,32);(11,12)(14,15)(17,18,20,22,21,24,27,29)(19,23,26,28,25,30,31,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(9,12,13,11)(10,15,16,14)(19,25)(23,30)(26,31)(28,32);(1,3,4,7)(2,6,5,8)(18,24)(22,29)(23,30)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256840, #72256846, #72256966, … |
| maximal normal subgroup of | #72256840, #72256846, #72256966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375489);