Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1926 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,26)(21,27)(22,31)(23,28)(25,29)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,28,25)(20,21,30,22)(26,27,32,31);(11,15)(13,16)(17,20,23,30)(18,25,28,24)(19,21,29,22)(26,27,32,31);(11,15)(13,16)(17,20)(18,26)(19,21)(22,29)(23,30)(24,27)(25,31)(28,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,22)(18,27,28,31)(19,20,29,30)(24,26,25,32)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308782, #72308886, #72308985, … |
| maximal normal subgroup of | #72308782, #72308886, #72308985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6379373);