Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(11,12)(14,16);(1,2,5,4)(3,6,8,7)(9,10,13,15)(11,16,12,14)(17,18,21,24)(19,20,23,22);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,24)(21,23);(1,3,5,8)(2,6,4,7)(9,10,13,15)(11,16,12,14)(17,20)(18,23)(19,24)(21,22);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,14,15,16)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256088, #72256306, #72262919, … |
| maximal normal subgroup of | #72256088, #72256306, #72262919, … |
| maximal quotient of | #72262673, #72300356, #72300841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004359);