Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115318 |
| 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 | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15,16,13)(12,14);(2,5)(3,8)(6,7)(13,15);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213195, #72213569, #72213597, … |
| maximal normal subgroup of | #72213195, #72213569, #72213597, … |
| maximal quotient of | C2×#115318, #72261287, #72261534 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22514);