Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360176 |
| 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 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,8)(6,7)(9,10,14,17)(11,18,19,24)(12,16,13,21)(15,22,23,20);(1,2)(3,6)(4,7)(5,8)(11,19)(15,23)(18,24)(20,22);(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(2,6)(3,8)(5,7);(2,7)(5,6)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231256, #72237138, C2×#6360336, … |
| maximal normal subgroup of | #72231256, #72237138, C2×#6360336, … |
| maximal quotient of | #72279390, #72279828, #72279938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267532);