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 | |
5778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,4,7,8)(9,10,14,15)(11,13,12,16)(18,20)(22,24);(1,3,6,7)(2,4,5,8)(10,15)(11,12)(19,23)(22,24);(1,4)(2,3)(5,7)(6,8)(9,11,14,12)(10,16,15,13)(17,18,21,20)(19,22,23,24);(1,5,6,2)(3,4,7,8)(9,12,14,11)(10,13,15,16)(17,19,21,23)(18,22,20,24);(9,13)(10,11)(12,15)(14,16)(17,20,21,18)(19,24,23,22);(1,4)(2,7)(3,5)(6,8)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=h,e^2=hi,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 | #72326547 |
| maximal normal subgroup of | #72326547 |
| maximal quotient of | #72326577, #72326753, #72327428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364040);