Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,14,17,15,12,13,18)(11,19,21,16,22,20,24,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,23)(13,24)(14,21)(15,22)(17,20)(18,19);(1,3)(2,5)(4,7)(6,8)(9,12)(10,15)(11,20)(13,18)(14,17)(16,21)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72216122, #72216126, #72216128, … |
| maximal normal subgroup of | #72216122, #72216126, #72216128, … |
| maximal quotient of | #72314163, #72314207 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22554);