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 | |
6453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,16)(14,15)(17,18,22,20)(19,24,21,23);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,23,20,24);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16)(18,20)(23,24);(3,7)(6,8)(11,16)(14,15)(17,20,22,18)(19,24,21,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21,22,19)(18,24,20,23);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,14,16,15)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72234848, #72234849, #72234868, … |
| maximal normal subgroup of | #72234848, #72234849, #72234868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7686652);