Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2137 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,32,29,31);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,21,22)(18,24,25,26)(19,28,30,29)(23,31,27,32);(10,14)(13,15)(19,29)(20,22)(23,32)(24,26)(27,31)(28,30);(2,5)(6,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=i,c^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239598, #72251200, #72256855, … |
| maximal normal subgroup of | #72239598, #72251200, #72256855, … |
| maximal quotient of | #72239676, #72239750, #72264831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7439191);