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 | |
6166 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)(3,4)(5,7)(6,8)(17,18,20,22)(19,21,23,24);(1,3,6,5)(2,7,8,4)(9,10)(11,14)(12,13)(15,16)(19,23)(21,24);(1,4,6,7)(2,5,8,3)(10,13)(14,16)(17,18)(19,21)(20,22)(23,24);(2,8)(3,5)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24);(1,5,6,3)(2,4,8,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72262893, #72262925, #72263013, … |
| maximal normal subgroup of | #72262893, #72262925, #72263013, … |
| maximal quotient of | #72263384, #72300748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8194982);