Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #393 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10)(11,17,20,23)(12,15)(13,16)(14,19,18,21)(22,24);(5,6,7,8)(9,11)(10,14)(12,21)(13,20)(15,23)(16,18)(17,24)(19,22);(1,2,3,4)(9,10)(11,18,20,14)(12,15)(13,16)(17,19,23,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234288, #72235146, #72235513, … |
| maximal normal subgroup of | #72234288, #72235146, #72235513, … |
| maximal quotient of | #72234302, #72235161 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31231);