Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(12,19)(14,20)(15,22)(18,23);(1,3)(2,5)(4,7)(6,8)(10,14,16,20)(15,21,22,24);(5,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24)Polycyclic presentation:
2^9|b^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233417, #72238457, #72238511, … |
| maximal normal subgroup of | #72233417, #72238457, #72238511, … |
| maximal quotient of | #72233391, #72238459, #72238460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161663);