Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3268 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,16)(11,17,18,24)(12,21,22,15)(14,19,23,20);(9,11,13,18)(10,14,16,23)(12,19,22,20)(15,24,21,17);(2,4)(11,18)(14,23)(17,24)(19,20);(11,19)(12,22)(14,24)(15,21)(17,23)(18,20);(5,7,8,6)(11,18)(14,23)(17,24)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72257400, #72257625, #72257730, … |
| maximal normal subgroup of | #72257400, #72257625, #72257730, … |
| maximal quotient of | #72264377, #72264450, #72264885 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291146);