Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,22)(21,25)(24,31)(26,29)(27,30)(28,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,30)(21,28)(22,27)(24,29)(25,32)(26,31);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,24)(19,26)(21,31)(22,25)(23,27)(28,30)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,25)(19,27)(22,24)(23,26)(28,29)(30,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,25)(20,31)(23,32)(26,30)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257261, #72257520, #72263999, … |
| maximal normal subgroup of | #72257261, #72257520, #72263999, … |
| maximal quotient of | #72264679 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496759);