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 | |
6797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(6,7)(10,15)(11,12)(17,18)(19,20)(21,23)(22,24);(1,2,5,3)(4,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,19,22,23)(18,20,24,21);(9,11,14,12)(10,16,15,13)(17,20,22,21)(18,23,24,19);(1,2,5,3)(4,7,8,6)(9,12,14,11)(10,16,15,13)(17,19)(18,20)(21,24)(22,23);(1,3,5,2)(4,7,8,6);(1,4,5,8)(2,6,3,7)(9,13,14,16)(10,12,15,11)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72234849, #72234868, #72234874, … |
| maximal normal subgroup of | #72234849, #72234868, #72234874, … |
| maximal quotient of | #72261318, #72261932, #72262210 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366139);