Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,13,16,14,18,12,17)(11,15,21,23,19,24,20,22);(5,6,7,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,23)(17,22)(18,24);(2,4)(10,16)(11,19)(12,13)(15,22)(17,18)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=ci,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234560, #72234566, #72234584, … |
| maximal normal subgroup of | #72234560, #72234566, #72234584, … |
| maximal quotient of | #72234564, #72279382, #72279868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53837);