Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,24,23)(18,22);(1,2)(3,4)(5,6,7,8)(9,11)(10,14)(12,17)(13,18)(15,23)(16,22)(19,21)(20,24);(1,3,4,2)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265630, #72265637, #72265648, … |
| maximal normal subgroup of | #72265630, #72265637, #72265648, … |
| maximal quotient of | #72266083, #72266101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30853);