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 | |
5701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,12)(14,16)(18,20)(22,23);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,20)(19,23,24,22);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,22,20,23);(9,10)(11,16)(12,14)(13,15)(17,20,21,18)(19,23,24,22);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256306, #72262978, #72262991, … |
| maximal normal subgroup of | #72256306, #72262978, #72262991, … |
| maximal quotient of | #72300761, #72300923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9696104);