Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^c=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72257778, #72302216, #72302217 |
| maximal normal subgroup of | #72257778, #72302216, #72302217 |
| maximal quotient of | #72257971 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92167);