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 | |
6628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,16,12,13);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,14)(15,16)(17,18,21,22)(19,23,24,20);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,13,12,16)(17,18)(19,23)(20,24)(21,22);(10,15)(11,12)(18,22)(19,24);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,13,15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72233573, #72233602, #72255837, … |
| maximal normal subgroup of | #72233573, #72233602, #72255837, … |
| maximal quotient of | #72261723, #72262796, #72263170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968304);