Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #59815 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(18,22)(19,20)(23,24);(2,5)(6,7)(10,13)(11,15)(14,16)(18,23)(19,20)(22,24);(9,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,23,20,22);(1,3,4,8)(2,6,5,7)(10,14)(13,16)Polycyclic presentation:
2^9|c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72257948, #72264318, #72264328 |
| maximal normal subgroup of | #72257948, #72264318, #72264328 |
| maximal quotient of | #72308798 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 166547);