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 | |
5885 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)(9,10,12,14)(11,15,16,13)(17,18,21,23)(19,20,24,22);(11,16)(13,15)(17,19,21,24)(18,22,23,20);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23);(1,3,4,7)(2,5,6,8)(9,10,12,14)(11,13,16,15)(17,20)(18,19)(21,22)(23,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72328083 |
| maximal normal subgroup of | #72328083 |
| maximal quotient of | #72328357, #72328842, #72329155 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315633);