Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1506 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)(11,19)(12,16,13,17)(14,18)(15,21,24,22)(20,23);(2,4)(5,6,7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24);(10,16)(11,20)(12,13)(15,19)(17,18)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=ci,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234560, #72234574, #72234578, … |
| maximal normal subgroup of | #72234560, #72234574, #72234578, … |
| maximal quotient of | #72234563, #72279237, #72279296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53914);