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 | |
5879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32);(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31);(2,5)(6,7)(10,14)(13,15)(18,23)(19,26)(24,28)(27,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,32)(29,30);(1,3,4,8)(2,6,5,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,28)(25,32)(26,29);(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326289, #72327271 |
| maximal normal subgroup of | #72326289, #72327271 |
| maximal quotient of | #72326291, #72326609, #72326723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7654517);