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 | |
6206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,16,14,12)(17,18,23,25)(19,22,27,24)(20,30,26,21)(28,29,32,31);(9,11)(10,12)(13,14)(15,16)(17,19)(18,24)(20,28)(21,29)(22,25)(23,27)(26,32)(30,31);(2,3)(6,7)(9,12,13,16)(10,14,15,11)(18,25)(19,27)(20,26)(29,31);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,20,23,26)(18,21,25,30)(19,28,27,32)(22,31,24,29);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,14)(15,16)(17,21)(18,26)(19,29)(20,25)(22,28)(23,30)(24,32)(27,31);(17,22)(18,27)(19,25)(20,31)(21,32)(23,24)(26,29)(28,30)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=ghi,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 | #72249479, #72249496, #72249517, … |
| maximal normal subgroup of | #72249479, #72249496, #72249517, … |
| maximal quotient of | #72300508, #72326692, #72326911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7812877);