Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,2,5,3,6,8,4,7)(9,11,12,20,13,19,21,18)(10,14,16,24,15,22,23,17);(1,3,5,8,6,7,4,2);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285487, #72286405, #72286509, … |
| maximal normal subgroup of | #72285487, #72286405, #72286509, … |
| maximal quotient of | #72285918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112266);