Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,29)(28,32);(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,3,4,8)(2,6,5,7)(11,16)(13,15);(10,14)(13,15)(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^9|b^2=f,d^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,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251246, #72256891, #72256902, … |
| maximal normal subgroup of | #72251246, #72256891, #72256902, … |
| maximal quotient of | #72307794 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6904051);