Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2053 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)(3,8)(4,6)(5,7)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(10,13)(14,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72281296, #72282600, #72282880 |
| maximal normal subgroup of | #72281296, #72282600, #72282880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6443227);