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 | |
7308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(9,10,12,14)(11,13,16,15)(17,18,21,23)(19,20,24,22);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,19,23,24);(17,19)(18,20)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72241089, #72241121, #72241128, … |
| maximal normal subgroup of | #72241089, #72241121, #72241128, … |
| maximal quotient of | #72328609, #72328877, #72329114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7582640);