Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,29)(27,30);(1,3,5,4)(2,6,8,7)(17,18,20,22)(19,26,25,32)(21,29,28,24)(23,30,31,27);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19,20,25)(18,23,22,31)(21,30,28,27)(24,32,29,26);(1,4,5,3)(2,6,8,7)(18,24)(21,28)(22,29)(23,32)(26,31)(27,30);(9,11,12,15)(10,14,13,16)(18,22)(23,31)(24,29)(26,32)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282648, #72282738, #72282893, … |
| maximal normal subgroup of | #72282648, #72282738, #72282893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364122);