Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22);(1,3,5,4)(2,6,8,7)(10,15)(11,13)(12,19)(16,18)(17,24)(20,21);(2,7)(3,4)(6,8)(9,11,14,13)(10,16,18,15)(12,19,20,21)(17,23,24,22);(9,12,14,20)(10,17,18,24)(11,19,13,21)(15,22,16,23);(10,18)(15,16)(17,24)(22,23)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339688, #72339713, #72339754 |
| maximal normal subgroup of | #72339688, #72339713, #72339754 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267146);