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 | |
6080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,16,14)(17,18)(19,26)(20,30)(21,27)(22,25)(23,24)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(17,19)(18,21)(20,28)(22,29)(23,27)(24,26)(25,31)(30,32);(2,4)(6,7)(9,11,13,16)(10,14,15,12)(18,24)(19,27)(20,25)(29,32);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,23,25)(18,22,24,30)(19,28,27,31)(21,29,26,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,19)(20,29)(22,28)(23,26)(24,27)(25,32)(30,31);(17,22)(18,25)(19,29)(20,24)(21,31)(23,30)(26,28)(27,32)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72235981, #72249484, #72249486, … |
| maximal normal subgroup of | #72235981, #72249484, #72249486, … |
| maximal quotient of | #72235748, #72235993, #72316590 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825909);