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 | |
2316 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)(9,10,13,18,14,17,23,19)(11,20,22,15,21,24,12,16);(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,20)(13,19)(14,17)(15,21)(18,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,22,14,21,23,12)(10,15,19,20,17,16,18,24);(3,8)(5,7)(9,12,13,11,14,22,23,21)(10,16,18,20,17,15,19,24);(2,6)(5,7)(10,17)(11,21)(12,22)(18,19)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=f,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, 6928559);