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 | |
5939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,20,25,28)(21,24,30,29)(26,27,31,32);(1,3)(2,6)(4,7)(5,8)(17,18)(19,20)(21,29)(22,23)(24,30)(25,28)(26,32)(27,31);(2,7)(4,6)(9,10,12,14)(11,13,16,15)(18,23)(19,25)(21,30)(27,32);(1,4,5,6)(2,3,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,20,23,28)(21,26,30,31)(24,27,29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32);(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249476, #72249505, #72249532, … |
| maximal normal subgroup of | #72249476, #72249505, #72249532, … |
| maximal quotient of | #72300535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7723245);