Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,15)(12,16,21,14)(13,17)(18,24)(19,23,20,22);(5,6,7,8)(10,14,17,16)(11,18)(15,23,24,22)(19,20);(2,4)(9,11,13,18)(10,15,17,24)(12,20,21,19)(14,22,16,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234964, #72234966, #72236937, … |
| maximal normal subgroup of | #72234964, #72234966, #72236937, … |
| maximal quotient of | #72236951 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53798);