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 | |
5940 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)(3,7)(4,8)(5,6)(11,15)(13,16)(21,29)(24,26)(27,32)(30,31);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,19)(23,25)(24,26)(30,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|b^2=hi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249491, #72249498, #72249524, … |
| maximal normal subgroup of | #72249491, #72249498, #72249524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074558);