Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,16,21,18)(11,19,12,15,22,23,24,20);(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,23)(13,18)(14,16)(15,22)(17,21)(19,24);(1,3)(2,5)(4,8)(6,7)(9,11,21,24,14,22,13,12)(10,15,17,23,16,20,18,19);(3,8)(5,7)(9,12,21,11,14,24,13,22)(10,15,18,19,16,20,17,23);(2,6)(5,7)(10,16)(11,22)(12,24)(17,18)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213818, #72219569 |
| maximal normal subgroup of | #72213818, #72219569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928563);