Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,26)(22,28)(23,27)(25,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,26)(21,30)(22,29)(23,25)(27,32)(28,31);(11,16)(13,15)(17,20,22,32,23,31,21,24)(18,25,26,30,27,19,28,29);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,25)(20,31)(24,32)(29,30);(2,5)(6,8)(17,21,23,22)(18,26,27,28)(19,29,25,30)(20,24,31,32)Polycyclic presentation:
2^9|c^2=g,d^2=fhi,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=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257477, #72257577, #72264267, … |
| maximal normal subgroup of | #72257477, #72257577, #72264267, … |
| maximal quotient of | #72307902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6692172);