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 | |
6062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,13,12)(11,15,16,14)(17,18,23,25)(19,21,20,24)(22,26,30,32)(27,29,28,31);(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,23)(21,25)(22,28)(26,29)(27,30)(31,32);(2,6)(3,7)(10,12)(14,15)(18,25)(19,20)(22,30)(29,31);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,23,19)(18,21,25,24)(22,28,30,27)(26,29,32,31);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,25)(22,29)(23,24)(26,27)(28,32)(30,31);(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249493, #72249505, #72249529, … |
| maximal normal subgroup of | #72249493, #72249505, #72249529, … |
| maximal quotient of | #72300573 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713837);