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 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,26)(22,24)(25,30)(27,32)(29,31);(1,2)(3,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,24,26)(20,27,30,29)(25,31,28,32);(2,5)(6,7)(9,11,12,16)(10,14,15,13)(18,24)(23,26)(25,28)(31,32);(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,28)(26,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21,22,19)(18,26,24,23)(20,29,30,27)(25,32,28,31)Polycyclic presentation:
2^9|b^2=h,c^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249465, #72249486, #72249544, … |
| maximal normal subgroup of | #72249465, #72249486, #72249544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7629949);