Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| 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×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,29)(28,32)(30,31);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,26,27)(20,30,22,28)(24,31,25,32);(9,11)(10,13)(12,16)(14,15)(18,24)(20,22)(23,31)(25,26)(27,32)(28,30);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72251246, #72251283, #72256872, … |
| maximal normal subgroup of | #72251246, #72251283, #72256872, … |
| maximal quotient of | #72264685, #72308413 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6909163);