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 | |
6464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,13)(14,16)(17,18,22,23)(19,26,25,21)(20,28,29,24)(27,30,31,32);(2,5)(6,7)(11,15)(14,16)(18,23)(19,25)(20,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,25)(24,28)(27,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,25)(18,21,23,26)(20,27,29,31)(24,30,28,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,31)(26,32);(17,21)(18,25)(19,23)(20,30)(22,26)(24,31)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249489, #72249501, #72249534, … |
| maximal normal subgroup of | #72249489, #72249501, #72249534, … |
| maximal quotient of | #72300500, #72326290, #72326608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019664);