Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2059 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,20)(23,24);(9,11)(10,15)(12,21)(13,23)(14,19)(16,20)(17,24)(18,22);(2,5)(3,8)(6,7)(11,19)(15,24)(18,20)(21,23);(9,10)(11,20)(12,16)(13,22)(14,17)(15,23)(18,19)(21,24);(9,12,14,13)(10,16,17,22)(11,21,19,23)(15,20,24,18)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72286234, #72286245, D8×#575, … |
| maximal normal subgroup of | #72286234, #72286245, D8×#575, … |
| maximal quotient of | #72319732, #72320666 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265452);